Perturbed Semi-Markov Type Processes II: Ergodic Theorems for Multi-Alternating Regenerative Processes [1st ed. 2022] 3030923983, 9783030923983

Citation preview

Dmitrii Silvestrov

Perturbed Semi-Markov Type Processes II Ergodic Theorems for Multi-Alternating Regenerative Processes

Perturbed Semi-Markov Type Processes II

Dmitrii Silvestrov

Perturbed Semi-Markov Type Processes II Ergodic Theorems for Multi-Alternating Regenerative Processes

123

Dmitrii Silvestrov Department of Mathematics Stockholm University Stockholm, Sweden

ISBN 978-3-030-92398-3 ISBN 978-3-030-92399-0 (eBook) https://doi.org/10.1007/978-3-030-92399-0 Mathematics Subject Classification: 60J10, 60J22, 60J27, 60K05, 60K15, 60K20, 65C40 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This book is the second volume of two-volume monograph devoted to the study of limit and ergodic theorems for regularly and singularly perturbed Markov chains, semi-Markov processes, and alternating regenerative processes with semi-Markov modulation. The second volume presents new super-long, long, and short time ergodic theorems for perturbed alternating regenerative processes and multi-alternating regenerative processes modulated by regularly and singularly perturbed finite semiMarkov processes. These theorems describe the asymptotic behaviour of distributions P{ξε (t) ∈ A} as time t → ∞, and the transition characteristics of perturbed multi-alternating regenerative processes ξε (t) converge to some limiting transition characteristics as the perturbation parameter ε → 0. Such “individual” ergodic theorems differ from ergodic theorems, such as the laws of large numbers for random averages and related ergodic theorems for the expectations of such averages, based on the use of stationarity arguments. The theorems presented in the book are based on the use of renewal type arguments in combination with new asymptotic recurrent algorithms of phase space reduction for perturbed multi-alternating regenerative processes. In the first volume, we present limit theorems on the weak convergence of distributions and the convergence of expectations of hitting times for regularly and singularly perturbed semi-Markov processes, based on algorithms of phase space reduction for perturbed semi-Markov processes. These results play an important role in obtaining the ergodic theorems presented in the second volume. Models of perturbed Markov chains and semi-Markov processes, in particular for the most complex cases of so-called singularly perturbed processes, attracted the attention of researchers in the middle of the twentieth century. Interest to these models has been stimulated by applications to control and queuing systems, information networks, epidemic models, and models of mathematical genetics and population dynamics. As a rule, Markov-type processes with singular perturbations arise as a natural tool for the mathematical analysis of multi-component systems with weakly interacting components. v

vi

Preface

The most deeply investigated are cases of Markov chains and semi-Markov processes with finite phase spaces. There is a huge bibliography of works that contain ergodic theorems and related limit theorems for hitting times for perturbed finite Markov chains and semi-Markov type processes. However, the theory of such ergodic and limit theorems is still far from completion. The results presented in the book, I hope, confirm this point of view well. The second volume includes an introduction, 12 chapters grouped in two parts, and three appendices. In Introduction (Chap. 1), simple examples, models of stochastic processes, conditions, and results are presented in an informal way. It also provides chapter-by-chapter content and additional information for potential readers. Part I (Chaps. 2–9) presents ergodic theorems for perturbed regenerative processes with regenerative lifetimes, as well as the results of a detailed analysis and complete classification of super-long, long, and short time ergodic theorems for regularly, singularly, or super-singularly perturbed alternating regenerative processes modulated by two-state semi-Markov processes. Part II (Chaps. 10–13) presents super-long, long, and short time ergodic theorems for regularly or singularly perturbed multi-alternating regenerative processes modulated by finite semi-Markov processes. This is done using new asymptotic recurrent algorithms for time-space aggregation of regenerative times and phase space reduction for modulating semi-Markov processes, which allow us to reduce the above ergodic theorems to the corresponding ergodic theorems for perturbed alternating regenerative processes presented in Part I. Appendix A presents generalisations of the classical renewal theorem to a model of the perturbed renewal equation. Appendix B presents some additional asymptotic results commonly used throughout the book. Appendix C contains some methodological and bibliographical notes, as well as comments on the new results presented in the book and some new problems for future research. I hope that the publication of this new book on asymptotic problems for perturbed stochastic processes will be a useful contribution to the ongoing intensive research in this area. In addition to being used for research and reference purposes, the book can also be used in special courses on this topic and as additional reading to general courses on stochastic processes. In this regard, it can be useful for both specialists as well as doctoral and senior students. I would also like to thank my colleagues at the Department of Mathematics, Stockholm University and at the Division of Mathematics and Physics, School of Education, Culture and Communication, Mälardalen University for creating an inspiring research environment and a friendly atmosphere, which stimulated my work. Stockholm, Sweden October 2021

Dmitrii Silvestrov

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Part I: Ergodic Theorems for Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Part I: Contents, Examples, Models, and Results . . . . . 1.1.2 Part I: Contents by Chapters . . . . . . . . . . . . . . . . . . . . . . 1.2 Part II: Ergodic Theorems for Perturbed Multi-Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Part II: Contents, Examples, Models, and Results . . . . 1.2.2 Part II: Contents by Chapters . . . . . . . . . . . . . . . . . . . . . 1.3 Appendices and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Appendix A: Perturbed Renewal Equation . . . . . . . . . . 1.3.2 Appendix B: Supplementary Asymptotic Results . . . . 1.3.3 Appendix C: Methodological and Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1

Part I

2

1 1 10 11 11 21 22 22 22 23 23

Ergodic Theorems for Perturbed Alternating Regenerative Processes

Ergodic Theorems for Perturbed Regenerative Processes . . . . . . . . . 2.1 Regenerative Processes with Regenerative Lifetimes . . . . . . . . . . 2.1.1 Regenerative Processes with Regenerative Lifetimes . . 2.1.2 Perturbation Conditions for Regenerative Processes with Regenerative Lifetimes . . . . . . . . . . . . . . . . . . . . . .

27 27 27 30

vii

viii

Contents

2.2

3

4

Ergodic Theorems for Perturbed Regenerative Processes with Regenerative Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Ergodic Theorems for Perturbed Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Ergodic Theorems for Perturbed Regenerative Processes with Modified Regenerative Lifetimes . . . . .

Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . 3.1 Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Alternating Regenerative Processes . . . . . . . . . . . . . . . . 3.1.2 Perturbation Conditions for Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Regularly, Singularly, and Super-Singularly Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Regular, Singular, and Super-Singular Perturbation Models for Alternating Regenerative Processes . . . . . . 3.2.2 Super-Long, Long, and Short Time Ergodic Theorems for Perturbed Alternating Regenerative Processes . . . . 3.3 Time Compression and Aggregation of Regeneration Times for Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . 3.3.1 Time Compression for Perturbed Regenerative and Alternating Regenerative Processes . . . . . . . . . . . . . . . . 3.3.2 Aggregation of Regeneration Times and Embedded Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Embedded Regenerative Processes and Ergodic Theorems for Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ergodic Theorems for Regularly Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Regularly Perturbed Alternating Regenerative Processes and Embedded Regenerative Processes of the First Type . . . . . . . . . . 4.1.1 Regularly and Semi-regularly Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Embedded Regenerative Processes of the First Type . . 4.2 Ergodic Theorems for Perturbed Standard Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Ergodic Theorems for Regularly Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Ergodic Theorems for Semi-regularly Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38 38 40 43 43 43 45 54 55 58 59 60 64

66 69 69 70 70 74 79 85

Contents

5

6

7

8

Ergodic Theorems for Regularly Perturbed Alternating Regenerative Processes Compressed in Time . . . . . . . . . . . . . . . . . . . . 5.1 Regularly Perturbed Alternating Regenerative Processes with Degenerating Regeneration Times . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Compression in Time for Regularly Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Super-Long and Long Time Ergodic Theorems for Singularly Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . 6.1 Singularly Perturbed Alternating Regenerative Processes and Aggregation of Regeneration Times . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Singularly Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Embedded Regenerative Processes of the Second Type 6.2 Super-Long Time Ergodic Theorems and Embedded Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Long Time Ergodic Theorems for Singularly Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . Short Time Ergodic Theorems for Singularly Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Short Time Ergodic Theorems Based on the First Time Compression Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Two Types of Time Compression Factors for Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . 7.1.2 Ergodic Theorems Based on the First Time Compression Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Short Time Ergodic Theorems Based on the Second Time Compression Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 First Type Short Time Ergodic Theorems Based on the Second Time Compression Factor . . . . . . . . . . . . . . . . . 7.2.2 Second Type Short Time Ergodic Theorems Based on the Second Time Compression Factor . . . . . . . . . . . . . . 7.2.3 Third Type Short Time Ergodic Theorems Based on the Second Time Compression Factor . . . . . . . . . . . . . . Ergodic Theorems for Singularly Perturbed Alternating Regenerative Processes Compressed in Time . . . . . . . . . . . . . . . . . . . . 8.1 Singularly Perturbed Alternating Regenerative Processes with Degenerating Regeneration Times . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Super-Long and Long Time Ergodic Theorems . . . . . . 8.1.2 Short Time Ergodic Theorems . . . . . . . . . . . . . . . . . . . . 8.2 Compression in Time for Singularly Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

93 93 101 107 107 107 108 123 128 135 135 135 136 139 139 141 146 149 149 149 155 157

x

9

Contents

Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Super-Long, Long, and Short Time Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes 9.1.1 Super-Singularly Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Super-Long Time Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Long Time Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes . . . . . . . 9.1.4 Short Time Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes . . . . . . . 9.2 Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes Compressed in Time . . . . . . . . . . . . . . . . 9.2.1 Super-Singularly Perturbed Alternating Regenerative Processes with Degenerating Regeneration Times . . . . 9.2.2 Compression in Time for Super-Singularly Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . 9.3 Generalisations and Classification of Ergodic Theorems for Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . 9.3.1 Generalisations of Ergodic Theorems for Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . . 9.3.2 Classification of Ergodic Theorems for Perturbed Alternating Regenerative Processes . . . . . . . . . . . . . . . .

163 164 164

165 167 169 171 172 172 173 174 180

Part II Ergodic Theorems for Perturbed Multi-Alternating Regenerative Processes 10

Perturbed Multi-Alternating Regenerative Processes . . . . . . . . . . . . . 10.1 Multi-Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . 10.1.1 Definition of Multi-Alternating Regenerative Processes 10.1.2 Perturbation Conditions for Multi-Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Multi-Alternating Regenerative Processes with Removed of Virtual Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Procedure of Total Removing of Virtual Transitions for Modulating Semi-Markov Processes . . . . . . . . . . . . . . . 10.2.2 Procedure of Partial Removing of Virtual Transitions for Modulating Semi-Markov Processes . . . . . . . . . . . . 10.3 Multi-Alternating Regenerative Processes with Reduced Modulating Semi-Markov Processes . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Procedure of One-State Reduction of Phase Space for Modulating Semi-Markov Processes . . . . . . . . . . . . . . . 10.3.2 Modified Procedure of One-State Reduction of Phase Space for Modulating Semi-Markov Processes . . . . . .

187 187 188 191 201 201 221 228 228 246

Contents

11

12

Time–Space Aggregation of Regeneration Times for Perturbed Multi-Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . 11.1 Multi-Alternating Regenerative Processes with Removed Virtual Transitions and Reduced Phase Space for Modulating Semi-Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Multi-Alternating Regenerative Processes with Totally or Partially Removed Virtual Transitions for Modulating Semi-Markov Processes . . . . . . . . . . . . . . . 11.1.2 Multi-Alternating Regenerative Processes with Reduced Phase Space of Modulating Semi-Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Time–Space Aggregation for Regeneration Times Based on Total Removing of Virtual Transitions . . . . . . . . . . . . . . . . . . . . . . 11.2.1 Algorithm of Time–Space Aggregation for Regeneration Times Based on Total Removing of Virtual Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Summary of Algorithm of Time–Space Aggregation for Regeneration Times Based on Total Removing of Virtual Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Time–Space Aggregation for Regeneration Times Based on Partial Removing of Virtual Transitions . . . . . . . . . . . . . . . . . . . . 11.3.1 Algorithm of Time–Space Aggregation for Regeneration Times Based on Partial Removing of Virtual Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Summary of Algorithm of Time–Space Aggregation of Regeneration Times Based on Partial Removing of Virtual Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Comparison of Recurrent Algorithms of Time–Space Aggregation for Regeneration Times . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Algorithms of Time–Space Aggregation of Regeneration Times Based on Total or Partial Removing of Virtual Transitions . . . . . . . . . . . . . . . . . . 11.4.2 Asymptotic Communicative Structure of Phase Spaces for Modulating Semi-Markov Processes for Embedded Alternating Regenerative Processes . . . . . . . . . . . . . . . . Embedded Processes for Perturbed Multi-Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Embedded Alternating Regenerative Processes . . . . . . . . . . . . . . 12.1.1 Embedded Alternating Regenerative Processes Based on Total Removing of Virtual Transitions . . . . . . . . . . . 12.1.2 Embedded Alternating Regenerative Processes Based on Partial Removing of Virtual Transitions . . . . . . . . .

xi

255

256

256

257 258

258

262 264

264

268 270

270

277 285 285 286 300

xii

Contents

12.2

13

Embedded Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Embedded Regenerative Processes Based on Total Removing of Virtual Transitions . . . . . . . . . . . . . . . . . . 12.2.2 Embedded Regenerative Processes Based on Partial Removing of Virtual Transitions . . . . . . . . . . . . . . . . . .

Ergodic Theorems for Perturbed Multi-Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Ergodic Theorems for Regularly Perturbed Multi-Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.1 Ergodic Theorems Based on Regularly Perturbed Embedded Alternating Regenerative Processes with Totally Removed Virtual Transitions . . . . . . . . . . . . . . . 13.1.2 Ergodic Theorems Based on Regularly Perturbed Embedded Alternating Regenerative Processes with Partially Removed Virtual Transitions . . . . . . . . . . . . . . 13.2 Ergodic Theorems for Singularly Perturbed Multi-Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Super-Long and Long Time Ergodic Theorems Based on Singularly Perturbed Embedded Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.2 Short Time Ergodic Theorems Based on Singularly Perturbed Embedded Alternating Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Ergodic Theorems for Perturbed Multi-Alternating Regenerative Processes Based on Embedded Regenerative Processes . . . . . . . 13.3.1 Ergodic Theorems Based on Perturbed Embedded Regenerative Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Relationship Between Ergodic Theorems Based on Perturbed Embedded Alternating Regenerative Processes and Embedded Regenerative Processes . . . .

A Perturbed Renewal Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 Renewal Equation and Renewal Theorem . . . . . . . . . . . . . . . . . . . A.1.1 Renewal Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.2 Non-Arithmetic Distribution Functions . . . . . . . . . . . . . A.1.3 Renewal Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Perturbed Renewal Equation and Renewal Theorem . . . . . . . . . . A.2.1 Perturbed Renewal Equation . . . . . . . . . . . . . . . . . . . . . A.2.2 Renewal Theorem for Perturbed Renewal Equation . . . A.2.3 Renewal Theorem for Perturbed Renewal Equation with Transition Renewal Period . . . . . . . . . . . . . . . . . . .

316 316 326 337 337

338

343 346

346

353 360 360

362 367 367 367 368 370 372 372 373 376

Contents

B

Supplementary Asymptotic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Limit Theorems for Stochastic Processes . . . . . . . . . . . . . . . . . . . B.1.1 Limit Theorems for Randomly Stopped Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.2 Slutsky Theorem and Related Results . . . . . . . . . . . . . . B.2 Other Useful Asymptotic Results . . . . . . . . . . . . . . . . . . . . . . . . . . B.2.1 Asymptotically Comparable Functions . . . . . . . . . . . . . B.2.2 Convergence of Lebesgue Integrals in the Scheme of Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

379 379 379 380 381 382 383

C Methodological and Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . . . C.1 Methodological Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 General Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . .

385 385 389

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

393

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

411

List of Symbols

⇒ d

−→ P

−→ a.s.

−→ J

−→ U

−→ us

−→ ∼ ≺ ε ε→0 d

= a.s.

Symbol of weak convergence Symbol of convergence in distribution Symbol of convergence in probability Symbol of almost sure convergence Symbol of convergence in topology J Symbol of convergence in topology U Symbol of locally uniform convergence Symbol of asymptotic equivalence Symbol of asymptotic dominance Perturbation parameter A shorter variant of symbol 0 < ε → 0 Symbol of equality for distributions

= ∗ (∗n) [u] k¯ n

Symbol of almost sure equality Symbol of convolution Symbol of n-fold convolution Integer part of a real number u Sequence of states k1, . . . , k n

ci,β

1 + β, if i = 1, or 1 + β−1 , if i = 2, for β ∈ [0, ∞] Interval [0, ∞) Borel σ-algebra of subsets of R+ Lebesgue measure on σ-algebra B+ Set of continuity points for a real-valued function f (·) defined on R+

R+ B+ m(A) C[ f (·)]

xv

xvi

List of Symbols

L

A, B, C, . . .

Class of real-valued Borel measurable functions defined on R+ and bounded in any finite interval A finite phase space {1, . . . , m} The σ-algebra of all subsets of the space X A phase space of a stochastic process (an arbitrary set) A σ-algebra of measurable subsets of the space Z A phase space of two-component stochastic process The σ-algebra of subsets C ⊆ Z of the form C = ∪i ∈B Ai × {i}, Ai ∈ BZ, i ∈ B ∈ BX . Class of non-negative functions q(t, A), which are Borel measurable in t ∈ R+ , for A ∈ BZ , and finite measures as functions of A ∈ BZ , for t ∈ R+ Conditions

ηε,n k ηε,n,

Embedded Markov chain Reduced embedded Markov chains

X BX Z BZ Z = Z × X BZ P[BZ ]

κε,n

k¯ n ηε,n, . . .

Transition time

ζε,n, ζˆε,n, ζˇε,n ηε (t) k ηε (t), k¯ n ηε (t), . . .

ξε (t) ξˆε (t), ξˇε (t) Q ε,i j (t) ei j (ε)

Moments of jumps / Regeneration times Semi-Markov process Reduced semi-Markov processes Regenerative process / Alternating regenerative process

k Q ε,i j (t), k¯ n Q ε,i j (t), . . .

Embedded regenerative processes Transition probability of semi-Markov process Expectation of transition time for semi-Markov processes Laplace transform of transition probability of semi-Markov process Transition probabilities for reduced

k eε,i j , k¯ n eε,i j , . . .

semi-Markov processes Expectations of transition time for reduced

k ψε,i j (s),

semi-Markov processes Laplace transform of transition probability

ψε,i j (s)

k¯ n ψε,i j (s), . . .

pε,i j k pε,i j , k¯ n pε,i j , . . .

for reduced semi-Markov processes Transition probability for embedded Markov chain Transition probabilities for reduced embedded Markov chains

List of Symbols

Fε,i j (t)

xvii

k Fε,i j (t), k¯ n Fε,i j (t), . . .

Conditional distribution function of transition time Conditional expectation of transition time Laplace transform for conditional distribution function of transition time Conditional distribution functions of transition

k fε,i j ,

times for reduced semi-Markov processes Conditional expectations of transition times

fε,i j φε,i j (s)

k¯ n fε,i j , . . .

k φε,i j (s), k¯ n φε,i j (s), . . .

for reduced semi-Markov processes Laplace transform for conditional distribution functions of transition times for reduced semi-Markov processes

(β)

(β)

π0, j (A), π0,i j (t, A), . . .

Limiting (stationary) probabilities

MC SMP MRP RP ARP MARP (*,*)1 , Lemma *1 , Theorem *1

Markov chain Semi-Markov process Markov renewal process Regenerative process Alternating regenerative process Multi-alternating regenerative process Forms of references to relations, lemmas, and theorems given in the first volume

Chapter 1

Introduction

This book is devoted to the study of ergodic theorems for regularly and singularly perturbed Markov chains, semi-Markov processes, and alternating regenerative processes with semi-Markov modulation. The introduction aims to informally present the main problems, methods, and algorithms that make up the content of the book. We give simple examples, illustrated by figures, and try to show the logic and ideas underlying the methods of asymptotic analysis of perturbed semi-Markov-type processes developed in the book as well as explain the meaning of the results presented in the book. We also describe the content of the book in parts and chapters and provide additional information for potential readers of the book.

1.1 Part I: Ergodic Theorems for Perturbed Alternating Regenerative Processes In Sect. 1.1, we outline the circle of problems and illustrate results presented in Part I.

1.1.1 Part I: Contents, Examples, Models, and Results 1.1.1.1 Content of Part I. In this part, which includes Chaps. 2–9, we present the results of the complete analysis of individual ergodic theorems for perturbed alternating regenerative processes modulated by regularly and singularly perturbed two-state semi-Markov processes. New short, long, and super-long time ergodic theorems for regularly, singularly, and super-singularly perturbed alternating regenerative processes are presented. A complete classification of ergodic theorems for perturbed alternating regenerative processes given in Part I is based on 26 such theorems.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0_1

1

2

1 Introduction

1.1.1.2 A Model Example. Let (ηε,n, κε,n ), n = 0, 1, . . ., be, for every ε ∈ (0, 1],1 a Markov renewal process that is a discrete time homogeneous Markov chain with the phase space Z = X × [0, ∞), where X = {1, 2}, and transition probabilities of the following form, for (i, s), ( j, t) ∈ Z, P{ηε,1 = j, κε,1 ≤ t/ηε,0 = i, κε,0 = s} = Fε,i j (t)pε,i j ,

(1.1)

where: (a) Fε,i j (t), t ≥ 0, is a distribution function without an atom in 0, for i, j ∈ X, and (b) pε,i j ≥ 0, i, j ∈ X, pε,i1 + pε,i2 = 1, i ∈ X. The transition probabilities of the Markov renewal process (ηε,n, κε,n ) depend only on the current state of the first component ηε,n . In this case, ηε,n is itself a discrete time homogeneous Markov chain, with phase space X and transition probabilities pε,i j , i, j ∈ X. It is so-called embedded Markov chain for the semi-Markov process ηε (t). We assume that pε,12, pε,21 > 0, ε ∈ (0, 1] and, thus, the Markov chain ηε,n is ergodic for every ε ∈ (0, 1]. The Markov renewal process (ηε,n, κε,n ) can be used to construct a semi-Markov process defined as ηε (t) = ηε,n for t ∈ [ζε,n, ζε,n+1 ), n = 0, 1, . . ., where ζε,n = n κ , n = 0, 1, . . .. The semi-Markov process ηε (t) has the phase space X and ε,k k=1 the transition probabilities Q ε,i j (t) = Fε,i j (t)pε,i j , t ≥ 0, i, j ∈ X. Let Sε be a queuing system, the functioning of which is described by the semiMarkov process ηε (t), t ≥ 0. The system Sε can operate in two, so-called work and failure, modes. If ηε (t) = 1, then the system is in the work mode at moment t. If ηε (t) = 2, then the system is in the failure mode at moment t. In this “queuing” interpretation, the random variable ηε,n is, for every n = 0, 1, . . ., a mode switching variable. It “decides” in which mode the system will operate in the functional period [ζε,n, ζε,n+1 ). Accordingly, the probabilities pε,i j have the meaning of the conditional mode switching probabilities. It should be mentioned that the probabilities pε,ii may take positive values. In such case, it is possible that the system may operate in the same mode during several consecutive functional periods [ζε,n, ζε,n+1 ). Figure 1.1 shows the transition graph for the embedded Markov chain ηε,n .

Fig. 1.1 The transition graph for the embedded Markov chain ηε, n

The random variable κε,n for each n = 0, 1, . . . is the duration of the n-th functional period. Respectively, Fε,i j (·) are conditional distribution functions of durations for functional periods. Let us consider one of the most simple alternating regenerative processes modulated by the semi-Markov process ηε (t): 1

Here and below, ε plays the role of a perturbation parameter.

1.1 Part I

3

κε (t) = t − ζε,n, for t ∈ [ζε,n, ζε,n+1 ), n = 0, 1, . . . .

(1.2)

The random variable κε (t) is the time between the last before t moment of regime switching and the moment t. Figure 1.2 shows a typical trajectory of the alternating regenerative process (κε (t), ηε (t)).

Fig. 1.2 A typical trajectory of the alternating regenerative process (κε (t), ηε (t))

Ergodic theorems are typical objects of interest in the study of perturbed queuing systems. One of the “ergodic” problems associated with the above alternating regenerative processes is the description of the asymptotic behaviour of the probabilities Pε,i j (t, Ax ) = Pi {κε (t) ≤ x, ηε (t) = j} as 0 ≤ t → ∞ and ε → 0, for Ax = [0, x], x ∈ [0, ∞), i, j ∈ X. Let us assume that the following perturbation conditions typical for ergodic theorems are satisfied for regime switching probabilities and distribution functions of durations for functional periods: (1) pε,i j → p0,i j as ε → 0,2 for i, j ∈ X, (2) Fε,i j (·) ⇒ F0,i j (·) as ε → 03 , for i, j ∈ X, where F0,i j (·) are non-arithmetic distribution functions, for i, j ∈ X such that p0,i j > 0. ∫∞ ∫∞ (3) fε,i j = 0 uFε,i j (du) → f0,i j = 0 uF0,i j (du) ∈ (0, ∞) as ε → 0, for i, j ∈ X. 2 3

In what follows, ε → 0 is a shorter version of the symbol 0 < ε → 0. ⇒ is the symbol of weak convergence for distribution functions.

4

1 Introduction

It turns out that asymptotics of the probabilities Pε,i j (t, Ax ) is largely determined by the asymptotics of the regime switching probabilities pε,12 and pε,21 , as ε → 0. This asymptotics essentially differs for cases, where the condition (1) is realised in one of the two basic variants of perturbation model. The first, (1a) pε,i j → p0,i j as ε → 0, for i  j, where p0,12 + p0,21 > 0, corresponds to the case of regularly perturbed queueing systems Sε . The second, (1b) pε,i j → p0,i j = 0 as ε → 0, for i  j, corresponds to the case of singularly perturbed queueing systems Sε . Also, the key role is played by the parameter βε = pε,12 /pε,21 and the condition: (4) βε → β ∈ [0, ∞] as ε → 0. This condition is automatically satisfied for regularly perturbed models but is an “independent” condition, additional to (1), for singularly perturbed models. The asymptotics of the probabilities Pε,i j (t, x) differs significantly for the cases, when: (4a) βε → β ∈ (0, ∞) as ε → 0, or (4b) βε → β as ε → 0, where β = 0 or β = ∞. Finally, the asymptotic behaviour of probabilities Pε,i j (tε, x) significantly differs for different asymptotic time zones determined by the following condition imposed on the time parameter 0 ≤ tε → ∞ as ε → 0: (5) tε /vε → t ∈ [0, ∞] as ε → 0, where the time compression factor vε is given by the following relation: −1 vε = p−1 ε,12 + pε,21 for ε ∈ (0, 1].

(1.3)

We refer to the ergodic theorems describing the asymptotic behaviour of the probabilities Pε,i j (tε, x) as super-long, long, or short time ergodic theorems in the cases, where, respectively, (5a) tε /vε → ∞, or (5b) tε /vε → t ∈ (0, ∞), or (5c) tε /vε → 0, as ε → 0. The method for obtaining ergodic relations for the probabilities Pε,i j (t, Ax ) is based on the construction of perturbed embedded regenerative processes for the processes (ηε (t), κε (t)) and the corresponding perturbed renewal equations for the above probabilities. This allows us to apply ergodic theorems for perturbed regenerative processes and variants of the renewal theorem for perturbed renewal equation presented in Chap. 2 and Appendix A. Various combinations of alternative variants for the conditions (1), (4), and (5) give a large number of different ergodic relations for the probabilities Pε,i j (tε, Ax ). Below, we present some of them: (I) This is the case of regularly perturbed systems, where the condition (1a) holds in the most simple variant, p0,12, p0,21 > 0. In this case, as mentioned above, the condition (4) is automatically satisfied, with the parameter β = p0,12 /p0,21 ∈ (0, ∞). −1 Also, the time compression factor vε → v0 = p−1 0,12 + p0,21 ∈ (0, ∞) as ε → 0, and, thus, the condition (5a) is automatically satisfied, for any 0 ≤ tε → ∞ as ε → 0. In this case, the limiting semi-Markov process η0 (t), with phase space X and transition probabilities Q0,i j (t) = F0,i j (t)p0,i j , t ≥ 0, i, j ∈ X, is ergodic and its stationary probabilities have the following form:

1.1 Part I

5

ρ0,1 = e0,1 (1 + β)−1 /e0, ρ0,2 = e0,2 (1 + β−1 )−1 /e0, (1.4)  where e0 = e0,1 (1 + β)−1 + e0,2 (1 + β−1 )−1 and e0,i = j ∈X f0,i j p0,i j , for i ∈ X. Further, the so-called steady transformation for the limiting distribution function  F0,i (·) = j ∈X F0,i j (·)p0,i j is defined by the following relation, for i ∈ X: −1 F¯0,i (x) = e0,i

∫

x

(1 − F0,i (u))du, x ≥ 0.

(1.5)

0

The following ergodic relation holds, for x ≥ 0, i, j ∈ X and any 0 ≤ tε → ∞ as ε → 0: (β) (1.6) Pε,i j (tε, Ax ) → π0, j (Ax ) = ρ0, j F¯0, j (x) as ε → 0. (II) The case of singularly perturbed systems, where the condition (1b) is satisfied, is more complicated. Note that, in this case, the time compression factor vε → ∞ ∈ (0, ∞) as ε → 0. Let us also assume that the condition (4a) is satisfied, with some parameter β ∈ (0, ∞). (IIa) In the case, where the condition (5a) is satisfied, the corresponding ergodic relation for the probabilities Pε,i j (tε, Ax ) takes the form of relation (1.6). (IIb) Let us assume that the condition (5b) is satisfied, with some value of parameter t ∈ (0, ∞). Let η(β) (t), t ≥ 0 be a continuous time homogeneous Markov chain with the −1 (1 + β), λ −1 −1 phase space X and transition intensities λ12 = e0,1 21 = e0,2 (1 + β ). Let, (β)

also, pi j (t) be transition probabilities for this Markov chain. Explicit expressions for (β)

the transition probabilities pi j (t) are well-known as solutions of the corresponding forward Kolmogorov system of differential equations. They are given in Sect. 6.3. The following ergodic relation holds, for x ≥ 0, i, j ∈ X and any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → t ∈ (0, ∞) as ε → 0: (β)

(β)

Pε,i j (tε, Ax ) → π0,i j (t, Ax ) = pi j (t)F¯0, j (x) as ε → 0.

(1.7)

(IIc) Finally, let us assume that the condition (5c) is satisfied. The following ergodic relation holds, for x ≥ 0, i, j ∈ X and any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → 0 as ε → 0: Pε,i j (tε, Ax ) → π¯0,i j (Ax ) = I( j = i)F¯0,i (x) as ε → 0.

(1.8)

It should be noted that the above ergodic relations can be also extended to a model, in which one of the limiting distribution functions F0,1 (·) or F0,2 (·) appearing in the condition (2) is concentrated at zero, if the parameter β appearing in the condition (5) takes a value, respectively, in the interval (0, ∞] or [0, ∞). (III) The case of singularly perturbed systems, where the condition (1b) is satisfied, becomes even more complicated if the condition (4b) is satisfied. In this case, another time compression factor also can be used: wε = (pε,12 + pε,21 )−1 for ε ∈ (0, 1].

(1.9)

6

1 Introduction

The condition (4b) implies that wε → ∞ and wε = o(vε ) as ε → 0. Let, for example, the parameter β = 0. The following ergodic relation holds, for x ≥ 0, i, j ∈ X and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → t ∈ (0, ∞) as ε → 0: (0) (0) ¯ Pε,i j (tε, Ax ) → π0,i j (t, Ax ) = pi j (t) F0, j (x) as ε → 0,

where p(0) i j (t)

⎧ ⎪ ⎪ ⎪ ⎨ ⎪

1 0 = (1 − e−t/e0,2 ) ⎪ ⎪ ⎪ ⎪ e−t/e0,2 ⎩

for for for for

j j j j

= 1, i = 2, i = 1, i = 2, i

= 1, = 1, = 2, = 2.

(1.10)

(1.11)

Note that p(0) i j (t) are transition probabilities for a continuous time, homogeneous Markov chain η(0) (t), t ≥ 0, with the phase space X and transition intensities λ12 = −1 . 0, λ21 = e0,2 The list of examples can be continued. The above examples represent very special cases of the results presented in Part I. This part presents the results of a complete asymptotic analysis of super-long, long, and short time ergodic theorems for perturbed alternating regenerative processes. We consider models of perturbed alternating processes with general regenerative components; study regularly, singularly and, also, super-singularly perturbed models; and expand ergodic theorems to more general models of perturbed alternating processes compressed in time. The classification, based on 26 ergodic theorems for perturbed alternating regenerative processes, is given in Chap. 9. 1.1.1.3 Regenerative Processes with Regenerative Lifetimes and Alternating Regenerative Processes. Two models of perturbed regenerative type processes are objects of the study in Part I. The first one is perturbed regenerative processes with regenerative lifetimes. The standard regenerative process is built using a sequence of “random blocks.” All blocks are independent and have the same probabilistic characteristics. The corresponding regenerative process is built by the sequential in time connection of blocks taken from the aforementioned sequence. Characteristic properties of the regenerative process ξε (t), t ≥ 0, are: (a) it forgets the past at the moment of regeneration (switching to a new random block) ζε,n , for any n = 0, 1, . . ., and (b) the time-shifted regenerating process ξε (ζε,n + t), t ≥ 0, is a probabilistic copy of the original regenerating process ξε (t), t ≥ 0, for any n = 0, 1, . . .. A random variable με defined on the same probability space as the process ξε (t), t ≥ 0, and possessing the loss-memory property at the regeneration moments ζε,n can be called regenerative lifetime. Figure 1.3 shows an example of the trajectory for a real-valued regenerative process ξε (t) with a regenerative lifetime με . The role of the regenerative lifetime με is played, in this case, by the first time of exceeding a level L by the regenerative process ξε (t).

1.1 Part I

7

Fig. 1.3 A regenerative process with regenerative lifetime

The standard alternating regenerative process is a natural generalisation of the regenerative process. It is built using sequences of “random blocks” of two types, say 1 and 2. Each block consists of a “piece” of a stochastic process of random duration. All blocks are independent. Blocks of each type have the same probabilistic characteristics. The corresponding alternating regenerative process is built by sequential alternate connection of blocks of types 1 and 2 taken from the above sequences. Part I explores more general alternating regenerative processes, in which a sequential alternate connection of blocks is controlled by some binary random variables. The “piece” of the stochastic process that creates each block, its duration, and the binary random variable that determines the decision to switch/not switch the block type at the end of the time interval corresponding to that block may be dependent. This allows us to speak about semi-Markov modulation for the corresponding alternating regenerative process. The above alternating regeneration process ξε (t), t ≥ 0, describes the functioning of some stochastic system. It is natural to interpret ξε (t) as the state of this system at moment t and the corresponding modulating semi-Markov process ηε (t) as a stochastic index, which shows that the system is in one of the two possible modes (for example, “works” or “does not work”) at the moment t if, respectively, ηε (t) = 1 or ηε (t) = 2. It is assumed that the joint probabilistic characteristics of the alternating regenerative process ξε (t) and the corresponding modulating semi-Markov process ηε (t), which controls the switching of block types, depend on some perturbation parameter ε ∈ (0, 1] and converge to the corresponding limiting joint characteristics, as ε → 0. Figure 1.4 below shows an example of the trajectory of an alternating regenerative process ξε (t) and the corresponding modulating semi-Markov index process ηε (t). We are interested in the so-called individual ergodic theorems describing the asymptotic behaviour of the joint distributions Pε,i j (t, A) = Pi {ξε (t) ∈ A, ηε (t) = j} for perturbed alternating or multi-alternating regenerative processes ξε (t) and modulating semi-Markov processes ηε (t), as time t → ∞ and the perturbation parameter ε → 0.

8

1 Introduction

Fig. 1.4 An alternating regenerative process

1.1.1.4 Perturbation Conditions for Alternating Regenerative Processes. The modulating semi-Markov process ηε (t) has the phase space X = {1, 2}. Let Q ε,i j (t) = Fε,i j (t)pε,i j be the transition probabilities of this process. We assume that the phase space X has the same communicative structure for ε ∈ (0, 1], that is, pε,i j > 0, ε ∈ (0, 1] or pε,i j = 0, ε ∈ (0, 1], for each i, j ∈ X. Models with three different types of perturbations are considered. These types are determined by the asymptotic behaviour of the transition probabilities pε,i j , i, j = 1, 2, as ε → 0, for the embedded Markov chain ηε,n of the semi-Markov process ηε (t). The first perturbation condition presupposes the convergence of the transition probabilities of embedded Markov chains: (A) pε,i j → p0,i j as ε → 0, for i, j ∈ X. The first class includes regularly and semi-regularly perturbed models, for which a Markov chain η0,n (with phase space X and transition probabilities p0,i j , i, j ∈ X) is ergodic that, in this case, is equivalent to the assumption, max(p0,12, p0,21 ) > 0. The second and third classes include singularly and super-singularly perturbed models, for which the Markov chain η0,n is not ergodic that is equivalent to the assumption, max(p0,12, p0,21 ) = 0. Two other basic perturbation conditions assume the weak convergence of the distribution functions of transition times: (B) Fε,i j (·) ⇒ F0,i j (·) as ε → 0, for i, j ∈ X, where F0,i j (·) is a non-arithmetic distribution function, for i, j ∈ X such that p0,i j > 0.

1.1 Part I

9

and the convergence of the first moments of the distribution functions Fε,i j (·) to the first moments of the corresponding limiting distribution functions: ∫∞ ∫∞ (C) fε,i j = 0 uFε,i j (du) → f0,i j = 0 uF0,i j (du) ∈ (0, ∞) as ε → 0, for i, j ∈ X. The fourth basic perturbation condition assumes the locally uniform convergence4 of functions qε,i (t, A) = Pi {ξε (t) ∈ A, ζε,1 > t} (which determine the free terms in the corresponding renewal type equations for the probabilities Pε,i j (t, A)), for A ∈ Γ, where Γ is a class of sets from the σ-algebra BZ of measurable subsets of the phase space Z (for the processes ξε (t)): us

(D) qε,i (·, A) −→ q0,i (·, A) as ε → 0 for s ∈ U A, A ∈ Γ, i ∈ X, where: (a) U A is some Borel subsets of [0, ∞), such that the Lebesgue measure m(U¯ A) = 0, (b) q0,i (t, A) is a measurable function continuous almost everywhere with respect to the Lebesgue measure on the Borel σ-algebra B+ of subsets of the interval [0, ∞), and (c) Γ ⊆ BZ and Z ∈ Γ. Note that the condition (D) implies that the class Γ can always be extended to its maximal form such that Γ would be closed with respect to the operations of the union of disjoint sets, the difference of sets connected by the relation of inclusion, and the complement. 1.1.1.5 Super-Long, Long, and Short Time Ergodic Theorems for Perturbed Alternating Regenerative Processes. The individual ergodic theorems for perturbed alternating regenerative processes modulating by regularly or singularly perturbed two-state semi-Markov processes are the main objects of the study in Part I. The most interesting are variants of the model with singular perturbations, where 0 < pε,12 → 0 as ε → 0 and 0 < pε,21 → 0 as ε → 0. In this case, an important role in shaping of limiting stationary distributions is played by the following balancing condition: (E) pε,12 /pε,21 → β ∈ [0, ∞] as ε → 0. Ergodic theorems take different forms in different asymptotic time zones determined by two time scaling factors: −1 −1 vε = p−1 → ∞ as ε → 0. ε,12 + pε,21 ≥ wε = (pε,12 + pε,21 )

(1.12)

Individual ergodic theorems for singularly perturbed models take the form of an asymptotic relation: (β)

Pε,i j (tε, A) → πi j (t, A) as ε → 0,

(1.13)

which holds for any 0 ≤ tε → ∞ as ε → 0 satisfying some time scaling relation: tε /vε → t ∈ [0, ∞] as ε → 0, or tε /wε → t ∈ [0, ∞] as ε → 0. us 4 −→

(1.14)

is the symbol of local uniform convergence in a point s for functions fε (·), which means that fε (sε ) → f0 (s) as ε → 0, for any sε → s as ε → 0. The details are given in Appendix B.

10

1 Introduction

These relations form asymptotically equivalent time zones, if β ∈ (0, ∞), but substantially different asymptotic time zones, if β = 0 or β = ∞. Indeed, the condition (E) implies that wε /vε → β/(1 + β2 ) as ε → 0. Obviously, wε = O(vε ) as ε → 0, if β ∈ (0, ∞), while wε = o(vε ) as ε → 0, if β = 0 or β = ∞. (β) The corresponding limiting probabilities πi j (t, A) may depend on parameter t and the initial state i of the modulating semi-Markov process, if t ∈ [0, ∞). These probabilities take substantially different forms, for cases t = ∞, t ∈ (0, ∞), and t = 0. We classify the corresponding theorems as super-long, long, and short time individual ergodic theorems, respectively.

1.1.2 Part I: Contents by Chapters Part I includes Chaps. 2–9. Chapter 2. In this chapter, we present ergodic theorems for perturbed regenerative processes with regenerative lifetimes. These theorems are direct corollaries of the corresponding versions of the renewal theorem for the perturbed renewal equation presented in Appendix A. They are the main tool for obtaining ergodic theorems for perturbed alternating regenerative processes. Chapter 3. In this chapter, the model of perturbed alternating regenerative processes is introduced, the main perturbation conditions are formulated, and general procedures of aggregation of regeneration times and time compression of regenerative and alternating regenerative processes are described. Chapter 4. In this chapter, we present ergodic theorems for regularly and semiregularly perturbed alternating regenerative processes. Chapter 5. In this chapter, we present ergodic theorems for regularly and semiregularly perturbed alternating regenerative processes compressed in time. Chapter 6. In this chapter, we present super-long and long time ergodic theorems for singularly perturbed alternating regenerative processes. Chapter 7. In this chapter, we present short time ergodic theorems for singularly perturbed alternating regenerative processes. Chapter 8. In this chapter, we present super-long, long, and short time ergodic theorems for singularly perturbed alternating regenerative processes compressed in time. Chapter 9. In this chapter, we present super-long, long, and short time ergodic theorems for super-singularly perturbed alternating regenerative processes. Also, the complete classification of ergodic theorems for perturbed alternating regenerative processes, which include 26 different variants of such theorems, is presented and commented.

1.2 Part II

11

1.2 Part II: Ergodic Theorems for Perturbed Multi-Alternating Regenerative Processes In Sect. 1.2, we outline the circle of problems and illustrate results presented in Part II.

1.2.1 Part II: Contents, Examples, Models, and Results 1.2.1.1 Content of Part II. The model of multi-alternating regenerative processes is a natural generalisation of the model of alternating regenerative processes described above. The difference is that such a process is built from a sequence of random blocks selected from a finite number of different types. Accordingly, it is modulated by a finite semi-Markov process. In Part II, which includes Chaps. 10– 13, we present asymptotic algorithms of time–space aggregation of regenerative times for perturbed multi-alternating regenerative processes. These algorithms are used to build for perturbed multi-alternating regenerative processes either embedded alternating regenerative processes or embedded regenerative processes. The above embedded processes let us apply ergodic theorems presented in Part I for obtaining super-long, long, and short time ergodic theorems for regularly and singularly perturbed multi-alternating regenerative processes. 1.2.1.2 A Model Example. Let ηε (t), t ≥ 0, be, for every ε ∈ (0, 1], a birth–death type semi-Markov process, which differs from the process introduced in Sect. 1.1.1.2, only by the phase space. It is, in this case, a three-state set X = {1, 2, 3}. As in Sect. 1.1.1.2, we assume that the transition probabilities for the semi-Markov process ηε (t) are of the form Q ε,i j (t) = Fε,i j (t)pε,i j , t ≥ 0, i, j ∈ X. Figure 1.5 shows the transition graph of the corresponding embedded Markov chain ηε,n . It is assumed that the probabilities pε,i j > 0, ε ∈ (0, 1] if |i − j | = 1, and pε,i j = 0, ε ∈ (0, 1] for transitions, which are not indicated by arrows.

Fig. 1.5 The transition graph for the embedded Markov chain ηε, n

Let Sε be a queuing system, the functioning of which is described by the semiMarkov process ηε (t), t ≥ 0. In this case, the system Sε can operate in three different, so-called work, partial work, and failure, modes.

12

1 Introduction

If ηε (t) = 1, then at the moment t the system is in the work mode. If ηε (t) = 2, then the system at the moment t is in the partial work mode. If ηε (t) = 3, then the system at the moment t is in the failure mode. In this case, we assume that the main perturbation conditions (1)–(3) are satisfied in the following modified forms: (6) pε,i j → p0,i j as ε → 0, for i, j ∈ X. (7) Fε,i j (·uε,i ) ⇒ F0,i j (·) as ε → 0, for i, j ∈ X, where (a) F0,i j (·) is non-arithmetic distribution function without singular component for i, j ∈ X such that po, ji > 0 and (b) uε,i ∈ (0, ∞) and uε,i → u0,i ∈ (0, ∞] as ε → 0, for i ∈ X. ∫ ∫∞ −1 ∞ (8) u−1 ε,i fε,i j = uε,i 0 uFε,i j (du) → f0,i j = 0 uF0,i j (du) ∈ (0, ∞) as ε → 0, for i, j ∈ X. In this example, some normalisation functions uε,i are used in the convergence relations (7) and (8). This explains the appropriate generalisation of the definition of the regenerative component as compared to those used in the example presented in Sect. 1.1.1.2. This component is now defined as κε (t) = u−1 ε,ηε, n (t − ζε,n ), for t ∈ [ζε,n, ζε,n+1 ), n = 0, 1, . . . .

(1.15)

The object of our interest is to describe the asymptotic behaviour of the probabilities Pε,i (t, Ax ) = Pi {κε (t) ≤ x} as 0 ≤ t → ∞ and ε → 0, for Ax = [0, x], x ∈ [0, ∞), i ∈ X. In what follows, we use the limits for various sums of products of the transition probabilities pε,i j and the normalisation functions uε,i and for the quotients of such sums and must guarantee the existence of such limits. Using the concept of a complete family of asymptotically comparable functions allows us to solve this problem. The family H of positive functions h(ε) defined on the interval (0, 1] is a complete family of asymptotically comparable functions if: (1) it is closed with respect to operations of summation, multiplication, and division, and (2) functions h(·) ∈ H have limits taking values in the interval [0, ∞], as ε → 0. The corresponding comments and examples of complete families of asymptotically comparable functions as well as operating rules and formulas for such functions are given in Chap. 81 . Also, short comments are given in Appendix B. Taking into account the above remarks, we also assume that the following condition is satisfied: (9) There exist a complete family of asymptotically comparable functions H such that: (a) for every i, j ∈ X, either pε,i j , ε ∈ (0, 1] belongs to the family H, or pε,i j = 0, ε ∈ (0, 1]; (b) uε,i, ε ∈ (0, 1] belongs to the family H, for i ∈ X. The analysis of the asymptotic behaviour of the probabilities Pε,i (t, Ax ) is based on the phase space reduction algorithms developed in Part II, which are applied to the alternating regenerative processes (κε (t), ηε (t)).

1.2 Part II

13

First, we must choose one of the least absorbing states. Suppose that the state 3 is such a state. This means that the following relation holds: lim

ε→0

p¯−1 ε,33 uε,3 p¯−1 ε,ii uε,i

= w0,3,i ∈ [0, ∞), for i ∈ X,

(1.16)

where p¯ε,ii = 1 − pε,ii, i ∈ X. Note that the condition (9) guarantees the existence of the above limits. Second, we must exclude the state 3 from the phase space X of the modulating semi-Markov process ηε (t). This can be done in two sub-steps. First, virtual transitions of the form 3 → 3 and 2 → 2 should be excluded from the trajectories of process ηε (t). In this way, the process ηε (t) will be transformed into a new semiMarkov process η˜ε (t). Second, “visits” to the state 3 should be excluded from the trajectories of the process η˜ε (t). In this way, the process η˜ε (t) will be transformed into a new semi-Markov process 3 ηε (t) with the reduced phase space 3 X = {1, 2}. Note that the state 1 is not involved in the above transformations, since a one-step transition from the state 1 to the state 3 is not possible. The regenerative component κε (t) does not change. The resulting vector process (κε (t), 3 ηε (t)) is a new alternating regenerative process with a modulating semiMarkov process 3 ηε (t). Figure 1.6 shows the transition graphs of the embedded Markov chains ηε,n , η˜ε,n , and 3 ηε,n for the semi-Markov processes, respectively, ηε (t), η˜ε (t), and 3 ηε (t).

Fig. 1.6 Transition graphs for the embedded Markov chains ηε, n , η˜ ε, n , and 3 ηε, n

Figure 1.7 shows typical trajectories of the processes ηε (t), η˜ε (t), 3 ηε (t) and the process κε (t). The difference in the slopes of the vectors on trajectory plot for the process κε (t) reflects the possible differences between normalisation functions uε,i, i ∈ X.

14

Fig. 1.7 Trajectories of processes ηε (t), η˜ ε (t), 3 ηε (t), and κε (t)

1 Introduction

1.2 Part II

15

Formulas for calculating the transition probabilities p˜ε,i j , i, j ∈ X, as functions of the initial transition probabilities pε,i j , i, j ∈ X, and the transition probabilities 3 pε,i j , i, j ∈ 3 X, as functions of the transition probabilities p˜ε,i j , i, j ∈ X, take the following forms: pε,i j for i = 1, j = 1, 2, 3, p˜ε,i j = (1.17) I( j  i)pε,i j / p¯ε,ii for i = 2, 3, j = 1, 2, 3, and 3 pε,i j

= p˜ε,i j + p˜ε,i3 p˜ε,3j for i, j = 1, 2.

(1.18)

Similar simple formulas for calculating the Laplace transforms and the first moments for the distribution functions of transition times for the semi-Markov processes η˜ε (t), as functions of the Laplace transforms and the first moments for the distribution functions of transition times for the semi-Markov processes ηε (t), and the Laplace transforms and the first moments for distribution functions of transition times for the semi-Markov processes 3 ηε (t), as functions of the Laplace transforms and the first moments for the distribution functions of transition times for the semi-Markov processes η˜ε (t), can be written. It is important that the above formulas for probabilities, Laplace transforms, and first moments let us prove that the perturbation conditions (6)–(8) are also satisfied for the semi-Markov processes η˜ε (t) and 3 ηε (t), with new normalisation functions, respectively, uε,1 for i = 1, (1.19) u˜ε,i = −1 p¯ε,ii uε,i, for i = 2, 3, and 3 uε,i

= u˜ε,i, for i = 1, 2.

(1.20)

Also, these formulas make it possible to calculate the corresponding limiting probabilities, Laplace transforms and first moments appearing in the conditions (6)–(8) for the semi-Markov processes η˜ε (t) and 3 ηε (t). The method for obtaining ergodic relations for the probabilities Pε,i (tε, Ax ) is based on the construction for the processes (κε (t), ηε (t)), either embedded alternating regenerative process with two-state modulating semi-Markov processes (such a case is presented in the above example) or embedded regenerative processes. This allows us to apply ergodic theorems for perturbed alternating regenerative processes and perturbed regenerative processes presented in Part I and variants of the renewal theorem for the perturbed renewal equations given in Appendix A. It should be noted that an important role in this asymptotic analysis is played by the functions qε,i (t, Ax ) = Pi {κε (t) ≤ x, ζε,1 > t}, t ≥ 0, which are free terms in the corresponding renewal type equations. We must assume or prove that the functions qε,i (tuε,i, Ax ) converge locally uniformly almost everywhere with respect to the Lebesgue measure to some limiting functions continuous almost everywhere with respect to the Lebesgue measure on B+ .

16

1 Introduction

In our example, functions qε,i (tuε,i, Ax ) = I(t ≤ x)(1 − Fε,i (tuε,i )), t ≥ 0, for i ∈ X, x ≥ 0. The condition (6) obviously implies that the above-mentioned locally uniform convergence takes place and the corresponding limiting functions are of the forms q0,i (t, Ax ) = I(t ≤ x)(1 − F0,i (t)), t ≥ 0, for i ∈ X, x ≥ 0. Moreover, there are explicit formulas for calculating the functions q˜ε,i (t, Ax ) and 3 qε,i (t, Ax ), which are analogues of the functions qε,i (t, Ax ), respectively, for the alternating regenerative processes (κε (t), η˜ε (t)) and (κε (t), 3 ηε (t)). These formulas allow us to prove that the above-mentioned relations of locally uniform convergence also hold for the functions q˜ε,i (t u˜ε,i, Ax ) and 3 qε,i (t 3 uε,i, Ax ) and calculate the corresponding limiting functions. Finally, it is easy to see that the normalisation functions u˜ε,i = 3 uε,i, i = 1, 2, can be replaced, in the corresponding asymptotic relations (7)–(8) for the Laplace transforms and the first moments of transition times for the semi-Markov processes 3 ηε (t) and the corresponding relation of locally uniform convergence for the functions 3 qε,i (t 3 uε,i, Ax ) by one normalisation function: uε = 3 uε,1 + 3 uε,2 .

(1.21)

The conditions (6)–(8) imply that the conditions (1)–(3) are satisfied for the compressed in time alternating regenerative processes (κε (tuε ), 3 ηε (tuε )) modulating by the two-state semi-Markov processes 3 ηε (t). The regenerative component κε (tuε ) is the same for the alternating regenerative processes (κε (tuε ), ηε (tuε )) and the alternating regenerative processes (κε (tuε ), 3 ηε (tuε )). This makes it possible to obtain ergodic relations for the probabilities Pε,i (tuε, Ax ) by applying the corresponding ergodic theorems for the alternating regenerative processes (κε (tuε ), 3 ηε (tuε )). As in the example considered in Sect. 1.1.1.2, there are a large number of variants of the corresponding ergodic relations. We restrict our presentation by one of the most interesting cases, where all states of the modulating semi-Markov processes ηε (t) are asymptotically absorbing, that is, the condition (6) is satisfied in the variant (6a) pε,i j → p0,i j as ε → 0, for i, j ∈ X, where the probabilities p0,ii = 1, for i ∈ X. From the condition (9), it follows that, in this case, the following asymptotic relation holds, which is an analogue of the condition (4): 3 pε,12 3 pε,21

=

pε,12 p¯ε,22 → 3 β ∈ [0, ∞] as ε → 0. pε,21

(1.22)

Also, the condition (9) implies that, the following asymptotic relation holds: 3 uε,2 3 uε,1

=

uε,2 → 3 γ ∈ [0, ∞] as ε → 0. uε,1 p¯ε,22

(1.23)

Both parameters 3 β and 3 γ can be calculated using operational rules for asymptotically comparable functions given in Chap. 81 . We restrict consideration by the case, where both parameters 3 β, 3 γ ∈ (0, ∞). The role of the additional time compression factor is played by the function:

1.2 Part II

17 3 vε

−1 = 3 p−1 ε,12 + 3 pε,21, ε ∈ (0, 1].

(1.24)

(IV) First, we present the corresponding super-long time ergodic relation for the case, where the following variant of the condition (5a), tε / 3 vε → ∞ as ε → 0, is satisfied. Let us denote (3 β, 3γ) 3 ρ0,i

=

⎧ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

−1 −1 3 e0,1 (1+ 3 γ) (1+ 3 β) 3 e0,2 (1+ 3

3 e0 γ −1 )−1 (1+

3

for i = 1,

β −1 )−1

3 e0

(1.25)

for i = 2,

where 3 e0 = 3 e0,1 (1 + 3 γ)−1 (1 + 3 β)−1 + 3 e0,2 (1 + 3 γ −1 )−1 (1 + 3 β−1 )−1 , 3 e0,1 = e0,1 ,  3 e0,2 = e0,2 + w0,3,2 e0,3 , and e0,i = j ∈X f0,i j p0,i j , for i ∈ X. The following ergodic relation holds, for x ≥ 0, i ∈ X, and any 0 ≤ tε → ∞ as ε → 0 such that tε / 3 vε → ∞ as ε → 0: ( β, 3γ)

Pε,i (tε uε, Ax ) → π03

( β, 3γ)

3 (Ax ) = 3 ρ0,1

( β, 3γ)

3 + 3 ρ0,2

3 π0,2 (Ax )

3 π0,1 (Ax )

as ε → 0,

(1.26)

where 3 π0, j (Ax ) is the limiting stationary distribution for the perturbed alternating regenerative processes (κε (t), 3 ηε (t)) with the reduced modulating semi-Markov processes 3 ηε (t). In addition, the asymptotic recurrent algorithm presented in Chap. 11 allows us to obtain the following explicit representation for the above limiting stationary distribution: for x ≥ 0, j = 1, F¯0,1 (x) π (A ) = (1.27) 3 0, j x e0,2 w0,3,2 e0,3 ¯ ¯ F0,2 (x) + F0,3 (x) for x ≥ 0, j = 2, e0,2 +w0,3,2 e0,3

e0,2 +w0,3,2 e0,3

∫ −1 x (1 − F (u))du, x ≥ 0, is the steady transformation for the where F¯0,i (x) = e0,i 0,i 0  limiting distribution function F0,i (·) = j ∈X F0,i j (·)p0,i j , for i ∈ X. (V) Second, let us present the corresponding long time ergodic relation for the case where the following variant of the condition (5b), tε / 3 vε → t ∈ (0, ∞) as ε → 0, is satisfied. Let η(3 β, 3γ) (t), t ≥ 0 be a continuous time homogeneous Markov chain with phase ( β, γ) −1 (1+ γ)(1+ β), λ (3 β, 3γ) = space 3 X = {1, 2} and transition intensities λ123 3 = 3 e0,1 3 3 21 −1 −1 −1 3 e0,2 (1 + 3 γ )(1 + 3 β ).

( β, 3γ)

Let, also, pi3j

(t) be the transition probabilities for this ( β, γ)

Markov chain. Explicit expressions for the transition probabilities pi3j 3 (t) are well-known as the solutions of the corresponding forward Kolmogorov system of differential equations. They are given in Sect. 6.3. The following ergodic relation holds, for x ≥ 0, i = 1, 2, and any 0 ≤ tε → ∞ as ε → 0 such that tε /3 vε → t ∈ (0, ∞) as ε → 0:

18

1 Introduction ( β, 3γ)

3 Pε,i (tε uε, Ax ) → π0,i

+

( β, 3γ)

(t, Ax ) = pi13

( β, γ) pi23 3 (t) 3 π0,2 (Ax )

(t) 3 π0,1 (Ax )

as ε → 0.

(1.28)

Also, it should be noted that, in this case, for x ≥ 0, Pε,3 (tε uε, Ax ) − Pε,2 (tε uε, Ax ) → 0 as ε → 0.

(1.29)

As in Sect. 1.1.1.2, the list of examples can be continued. The above examples present results that are very special cases of the results presented in Part II. In this part, asymptotic recurrent algorithms of phase space reductions for regularly and singularly perturbed multi-alternating regenerative processes with general regenerative component are presented; procedures for constructing embedded alternating regenerative processes and embedded regenerative processes are described; and super-long, long, and short time ergodic theorems for perturbed multi-alternating regenerative processes are obtained. 1.2.1.3 Multi-Alternating Regenerative Processes and Algorithms of Phase Space Reduction. In Part II, we present ergodic theorems for perturbed multialternating regenerative processes modulated by regularly or singularly perturbed finite semi-Markov processes. These processes are constructed in the manner described in Sect. 1.1.1.3 for alternating regenerative processes. In this case, the multi-alternating regenerative process (ξε (t), ηε (t)) has a modulating semi-Markov process ηε (t) with a finite phase space X = {1, . . . , m}. The initial characteristics are the transition probabilities Q ε,i j (t) = Fε,i j (t)pε,i j for the modulating semi-Markov processes ηε (t) and the probabilities qε,i (t, A) = P{ξε (t) ∈ A, ζε,1 > t}, which play the role of free terms in the corresponding renewal type equations for the probabilities Pε,i (t, A) = Pi {ξε (t) ∈ A, ηε (t) = j}, t ≥ 0, i, j ∈ X. As in Part I, we are interested in the so-called individual ergodic theorems describing the asymptotic behaviour of the joint distributions Pε,i j (t, A) = Pi {ξε (t) ∈ A, ηε (t) = j}, as time t → ∞ and the perturbation parameter ε → 0. As above, we assume the following standard regularity conditions for the perturbed semi-Markov processes ηε (t), namely: (a) the communicative structure of the phase space X is the same for all ε ∈ (0, 1], that is, either pε,i j > 0, ε ∈ (0, 1], or pε,i j = 0, ε ∈ (0, 1], for each i, j ∈ X and (b) X is one class of communicative states, for ε ∈ (0, 1], (c) Fε,i j (0) = 0, ε ∈ (0, 1], for i, j ∈ X. In Part II, we develop new asymptotic recurrent algorithms for aggregating the regeneration times and reducing the phase space for modulating semi-Markov processes. These algorithms allow embedding models with a finite phase space of modulating semi-Markov processes in models with two-state modulating semi-Markov processes. At each step of the above recurrent algorithm, one of the least absorbing states k n is excluded from the phase space of the corresponding semi-Markov process k¯ n−1 ηε (t), with the phase space k¯ n−1 X = X \ {k1, . . . , k n−1 }. New semi-Markov process k¯ n ηε (t), with the phase space k¯ n X = X \ {k1, . . . , k n }, is constructed. The moments of jumps for this process are the successive moments of hitting (as results of jumps) in the phase space k¯ n X by the semi-Markov process k¯ n−1 ηε (t). The states of both processes

1.2 Part II k¯ n−1 ηε (t) and k¯ n ηε (t) coincide ξε (t) does not change.

19

at these moments. The first regenerative component

Figures 1.8 illustrates this algorithm. In this case, as in the example considered in Sect. 2.1.1.2, the regenerative component is defined as in the relation (1.15).

Fig. 1.8 Reduction of phase space for modulating semi-Markov process

20

1 Introduction

Two states k1 and k2 are successively excluded from the initial phase space X = {k1, k2, k3, k4 }. The resulting alternating regenerative process with semiMarkov modulation has the same first regenerative component (as the original multi-alternating regenerative process) and new regeneration times, which are the successive moments of hitting the two-state subset k¯2 X = {k3, k4 } by the original modulating semi-Markov process. The states k3 and k4 are the most absorbing states for the initial modulating semi-Markov process. Figure 1.9 also illustrates this algorithm. In Fig. 1.9, these two the most absorbing states are shown in blue. The new reduced modulating semi-Markov process has a transition period, which ends at the moment of first hitting in the above two-state subset by the original modulating semi-Markov process.

Fig. 1.9 Embedding procedure for multi-alternating regenerative processes

The corresponding transitions are indicated by black arrows and the symbol 1. Transitions in the first time interval between the moments of hitting in one of the “blue” states are indicated by blue arrows and the symbol 2, transitions in the second time interval between the moments of hitting in one of the “blue” states are indicated by blue arrows and the symbol 3, etc. Since the regenerative component is the same for the original multi-alternating regenerative process and the reduced alternating regenerative process, the ergodic theorems for the above reduced perturbed alternating processes modulated by twostate semi-Markov processes allow us to obtain ergodic theorems for the original perturbed multi-alternating processes modulated by regularly or singularly perturbed finite semi-Markov processes. 1.2.1.4 Perturbation Conditions and Ergodic Theorems for Perturbed MultiAlternating Regenerative Processes. The aggregation of regenerative times can

1.2 Part II

21

lead to the need to compress the time for new multi-alternating regenerative processes with reduced modulating semi-Markov processes. However, we find the balanced forms of initial perturbation conditions analogous to the conditions (A)–(D) such that the corresponding multi-alternating regenerative processes with reduced modulating semi-Markov processes inherit these perturbation conditions. These conditions are: (A) pε,i j → p0,i j as ε → 0, for i, j = 1, 2. (B) Fε,i j (·uε,i ) ⇒ F0,i j (·) as ε → 0, for i, j ∈ X, where (a) F0,i j (·) are non-arithmetic distribution functions without singular component, for i, j ∈ X such that p0,i j > 0 and (b) uε,i ∈ (0, ∞) and uε,i → u0,i ∈ (0, ∞] as ε → 0, for i ∈ X. ∫∞ ∫∞ (C) u−1 ε,i 0 uFε,i j (du) → 0 uF0,i j (du) ∈ (0, ∞) as ε → 0, for i, j ∈ X. us

(D) qε,i (·uε,i, A) −→ q0,i (·, A) as ε → 0, for s ∈ U A, A ∈ Γ, i ∈ X, where: (a) U A is some Borel subset of [0, ∞) such that the Lebesgue measure m(U¯ A) = 0, (b) q0,i (t, A) is some measurable function continuous almost everywhere with respect to the Lebesgue measure on B+ , (c) Γ ⊆ BZ and Z ∈ Γ. The following condition is also used, similar to the condition (9): (F ) There exists a complete family of asymptotically comparable functions H such that: (a) pε,i j , ε ∈ (0, 1] belongs to the family H, for i, j ∈ X such that pε,i j > 0, ε ∈ (0, 1] and (b) uε,i, ε ∈ (0, 1] belongs to the family H, for i ∈ X. We also provide explicit recurrent formulas for re-calculating the corresponding time compression factors and limits for transition characteristics of multi-alternating regenerative processes with reduced modulating semi-Markov processes. The above asymptotic recurrent algorithms of phase space reduction are used for aggregation of regeneration times, construction of embedded alternating regenerative processes and embedded regenerative processes, and obtaining ergodic theorems for regularly perturbed multi-alternating regenerative processes and super-long, long, and short time ergodic theorems for singularly perturbed multi-alternating regenerative processes. Interestingly, super-long time ergodic theorems based on the use of embedded regenerative or alternating regenerative processes are equivalent, except some minor differences. While long and short time ergodic theorems for singularly perturbed multi-alternating regenerative processes obtained with the use of embedded alternating regenerative processes are stronger than the corresponding theorems obtained with the use of embedded regenerative processes.

1.2.2 Part II: Contents by Chapters Part II includes Chaps. 10–13. Chapter 10. In this chapter, we introduce a model of perturbed multi-alternating regenerative processes modulating by regularly or singularly perturbed finite semi-

22

1 Introduction

Markov processes. The corresponding perturbation conditions are formulated. Also, procedures of removing of virtual transitions and one-state reduction of phase space for modulating semi-Markov processes and related procedures for multi-alternating regenerative processes are described. Chapter 11. In this chapter, we describe asymptotic recurrent algorithms of phase space reduction for multi-alternating regenerative processes. Here, the asymptotic recurrent algorithms of phase space reduction for perturbed semi-Markov processes, presented in the first volume, are essentially used. Chapter 12. In this chapter, we describe procedures for constructing embedded alternating regenerative processes and embedded regenerative processes for perturbed multi-alternating regenerative processes and formulate the corresponding perturbation conditions for these embedded processes. Chapter 13. In this chapter, we present ergodic theorems for perturbed multialternating regenerative processes modulating by regularly or singularly perturbed finite semi-Markov processes. These theorems are obtained using algorithms of time– space aggregation of regeneration times based on the reduction of the phase space for modulating semi-Markov processes and ergodic theorems for the corresponding perturbed embedded alternating regenerative processes and embedded regenerative processes for perturbed multi-alternating regenerative processes presented in Part I. Here, an essential role is played by the limit theorems for distributions of hitting times and their expectations for regularly and singularly perturbed semi-Markov processes presented in Volume 1.

1.3 Appendices and Conclusion 1.3.1 Appendix A: Perturbed Renewal Equation In this appendix, we present the results connected with generalisation of the classical renewal theorem to the model of perturbed renewal equation. These theorems are one of the main tools used for obtaining ergodic theorems for perturbed alternating and multi-alternating regenerative processes modulating by regularly or singularly perturbed semi-Markov processes.

1.3.2 Appendix B: Supplementary Asymptotic Results In this appendix, we present some asymptotic results often used in the book.

1.3 Appendices and conclusion

23

1.3.3 Appendix C: Methodological and Bibliographical Notes This appendix includes methodological and historical notes connected with results presented in the book, comments concerning new results presented in the book and related new problems for future research, and bibliographical notes to the bibliography of works in the area of ergodic theorems for perturbed Markov-type stochastic processes. It supplements the bibliography of works in the area of asymptotic problems for perturbed Markov type stochastic processes and systems, given the first volume of the monograph.

1.3.4 Conclusion The asymptotic analysis for perturbed processes of Markov type and related problems are subjects of intensive studies during several decades. However, the development of the theory, in particular, its parts connected with computational algorithms for singularly perturbed models as well as ergodic theorems for singularly perturbed Markov type processes, is still far from the completion. The book is concentrated in this area. New computational asymptotic algorithms of phase space reduction presented in the book are universal. They can be applied to perturbed multi-alternating regenerative processes modulating by perturbed finite semi-Markov processes with an arbitrary asymptotic communicative structure of phase spaces. The corresponding algorithms are computationally effective due to recurrent character of computational procedures. Ergodic theorems for perturbed multi-alternating regenerative processes modulating by singularly perturbed semi-Markov processes presented in the book were not known before. The results presented in the book will be interesting to specialists, who work in such areas of the theory of stochastic processes as ergodic, limit, and large deviation theorems, analytical and computational methods for Markov chains, Markov, semiMarkov, regenerative and other classes of Markov type stochastic processes and their queuing, network, bio-stochastic and other applications. I hope that the book will also attract attention of those researchers, who are interested in new analytical methods of analysis for nonlinearly perturbed stochastic processes and systems. It can also be useful for doctoral and senior students. This gives me hope that the book will find sufficient number of readers interested in stochastic processes and their applications.

Part I

Ergodic Theorems for Perturbed Alternating Regenerative Processes

Chapter 2

Ergodic Theorems for Perturbed Regenerative Processes

In this chapter, we introduce a model of perturbed regenerative processes with regenerative lifetimes and present ergodic theorems for such processes. This chapter includes two sections. In Sect. 2.1, we introduce a model of perturbed regenerative processes with regenerative lifetimes and formulate perturbation conditions for such processes. In Sect. 2.2 we present ergodic theorems for perturbed regenerative processes with regenerative lifetimes, which are one of the main analytical tools in our studies of ergodic theorems for perturbed alternating and multi-alternating regenerative processes. The corresponding asymptotic results are stated in Theorems 2.1–2.3.

2.1 Regenerative Processes with Regenerative Lifetimes In this section, we introduce the model of perturbed regenerative processes with regenerative lifetimes and formulate perturbation conditions for such processes.

2.1.1 Regenerative Processes with Regenerative Lifetimes 2.1.1.1 Regenerative Processes and Regenerative Lifetimes. Let Ωε, Fε, Pε be, for every ε ∈ (0, 1], a probability space. We assume that all stochastic processes and random variables introduced below and indexed by the parameter ε are defined on the probability space Ωε, Fε, Pε . Let us assume that the following model assumptions are fulfilled: (A) ξ¯ε,n = ξε,n (t), t ≥ 0 be, for every n = 1, 2, . . ., a stochastic process with a phase space Z (with the corresponding σ-algebra of measurable subsets BZ ), measurable in the sense that ξε,n (t, ω), (t, ω) ∈ [0, ∞) × Ωε is a measurable function of (t, ω) (this means that {(t, ω) ∈ A} ∈ B+ ⊗ Fε , for A ∈ BZ , where B+ ⊗ Fε is the minimal σ-algebra containing all sets B × C, for B ∈ B+, C ∈ Fε . © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0_2

27

28

2 Ergodic theorems for perturbed RP

(B) κε,n be, for every n = 1, 2, . . ., a non-negative random variable. (C) με,n be, for every n = 1, 2, . . ., a non-negative random variable. (D) Stochastic triplets ξ¯ε,n = ξε,n (t), t ≥ 0 , κε,n, με,n , n = 1, 2, . . ., are mutually independent. (E) Joint distributions of random variables ξε,n (tk ), k = 1, . . . , r, and κε,n, με,n do not depend on n ≥ 1, for every tk ∈ [0, ∞), k = 1, . . . , r, r ≥ 1. Let us define a standard regenerative process: ξε (t) = ξε,n (t − ζε,n−1 ), for t ∈ [ζε,n−1, ζε,n ), n = 1, 2, . . . ,

(2.1)

with regeneration times: ζε,n = κε,1 + · · · + κε,n, n = 1, 2, . . . , ζε,0 = 0,

(2.2)

and a regenerative lifetime: με =

ν

ε −1

κε,k + με,νε I(νε < ∞),

(2.3)

k=1

where νε = min(n ≥ 1 : με,n < κε,n ).

(2.4)

Note that νε and με may be proper or improper random variables. We assume that the following condition holds: O1 : (a) P{κε,1 = 0} = 0, for ε ∈ (0, 1], (b) P{με,1 < κε,1 } ∈ [0, 1), for ε ∈ (0, 1]. The condition O1 (a) implies that, for every ε ∈ (0, 1], the random variables P

ζε,n −→ ∞ as n → ∞. Therefore, the process ξε (t) is well-defined on the time interval [0, ∞). The condition O1 (b) excludes, for every ε ∈ (0, 1], the degenerated case, where με = με,1 < ζε,1 , with probability 1, and, thus, νε = 1, with probability 1. The object of our interest is probabilities: Pε (t, A) = P{ξε (t) ∈ A, με > t}, A ∈ BZ, t ≥ 0.

(2.5)

Let L be a class of real-valued Borel measurable functions defined on R+ and bounded on any finite interval. Obviously, Pε (t, A) ∈ [0, 1], A ∈ BZ, t ≥ 0. Also, the function Pε (t, A), t ≥ 0, belongs, for every A ∈ BZ , to the class L and is the unique in this class solution for the following renewal equation: ∫ t Pε (t, A) = qε (t, A) + Pε (t − s, A)Fε (ds), t ≥ 0, (2.6) 0

where

2.1 RP with regenerative lifetimes

29

Fε (t) = P{ζε,1 ≤ t, με ≥ ζε,1 } = P{κε,1 ≤ t, με,1 ≥ κε,1 }, t ≥ 0,

(2.7)

qε (t, A) = P{ξε (t) ∈ A, ζε,1 ∧ με > t} = P{ξε,1 (t) ∈ A, κε,1 ∧ με,1 > t}, A ∈ BZ, t ≥ 0.

(2.8)

and

Note also that Fε (t) is a proper or improper distribution function on interval [0, ∞). Also, qε (t, A) ∈ [0, 1], t ≥ 0, A ∈ BZ , and the function qε (t, A), t ≥ 0, belongs to the class L, for every A ∈ BZ . 2.1.1.2 Regenerative Processes with Regenerative Lifetimes and Transition Period. Let us now assume that the model assumption (E) formulated in Sect. 2.1.1.1 holds only for n ≥ 2, i.e., takes the following form: ¯ Joint distributions of random variables ξε,n (tk ), k = 1, . . . , r, and κε,n, με,n (E) do not depend on n ≥ 2, for every tk ∈ [0, ∞), k = 1, . . . , r, r ≥ 1. In this case, the process ξε (t), t ≥ 0, is usually called the regenerative process with a transition period [0, ζε,1 ). In this case, the shifted process: ξε(1) (t) = ξε (ζε,1 + t), t ≥ 0

(2.9)

is a standard regenerative process with regeneration times: (1) (1) ζε,n = κε,2 + · · · + κε,n+1, n = 1, 2, . . . , ζε,0 = 0,

(2.10)

and the corresponding shifted regenerative lifetime is defined as (1)

μ(1) ε

=

ν

ε −1 k=1

where

κε,1+k + με,1+ν (1) I(νε(1) < ∞), ε

νε(1) = min(n ≥ 1 : με,1+n < κε,1+n ).

(2.11)

(2.12)

All quantities appearing in the renewal equation (2.6) and relations (2.8) and (1) (2.7) should be defined using a shifted sequence of triplets ξ¯ε,n = ξε,n+1 (t), t ≥ 0 , κε,n+1, με,n+1 , n = 1, 2, . . .. It is also natural to index the above-mentioned quantities by the upper index (1) , for example, to use the notation Pε(1) (t, A) = P{ξε(1) (t) ∈ (1) A, μ(1) ε > t}, etc. The probabilities Pε (t, A) satisfy the renewal equation (2.6). In this case, Theorem 2.1 (formulated below) presents the corresponding ergodic relation for these probabilities. The probabilities Pε (t, A) = P{ξε (t) ∈ A, με > t}, defined for the original regenerative process with transition period, and the probabilities Pε(1) (tε, A) are related by

30

2 Ergodic theorems for perturbed RP

the following renewal type transition relation: ∫ t Pε (t, A) = qε (t, A) + Pε(1) (t − s, A)Fε (ds), t ≥ 0,

(2.13)

0

where, for t ≥ 0,

Fε (t) = P{ζε,1 ≤ t, με,1 ≥ ζε,1 }, t ≥ 0.

(2.14)

and, for A ∈ BZ, t ≥ 0, qε (t, A) = P{ξε (t) ∈ A, ζε,1 ∧ με > t} = P{ξε,1 (t) ∈ A, ζε,1 ∧ με,1 > t}.

(2.15)

Note that Fε (t) can be a proper or improper distribution function on interval [0, ∞). It is also possible that the transition period can be of zero duration and, thus, the distribution function Fε (t) can possess an atom at zero or even be concentrated at zero, for ε ∈ (0, 1]. Also qε (t, A) ∈ [0, 1], A ∈ BZ, t ≥ 0, and the function qε (t, A), t ≥ 0, belongs to the class L, for every A ∈ BZ .

2.1.2 Perturbation Conditions for Regenerative Processes with Regenerative Lifetimes 2.1.2.1 Let us introduce the distribution functions, for ε ∈ (0, 1]: F¯ε (t) = P{ζε,1 ≤ t} = P{κε,1 ≤ t}, t ≥ 0,

(2.16)

and stopping probabilities: qε = 1 − Fε (∞) = P{με < ζε,1 } = P{με,1 < κε,1 }.

(2.17)

If qε > 0, then, P{νε < ∞} = P{με < ∞} = 1. If qε = 0, then, P{νε = ∞} = P{με = ∞} = 1. We assume that the following condition is satisfied: P1 : (a) F¯ε (·) ⇒ F¯0 (·) as ε → 0, where F¯0 (u) is a proper distribution function concentrated on interval [0, ∞), (b) F¯0 (u) is a weakly non-arithmetic distribution function, and (c) qε → q0 = 0 as ε → 0. Lemma 2.1 If the condition P1 (c) is satisfied, then the condition P1 (a) is equivalent to the following weak convergence relation: Fε (·) ⇒ F0 (·) ≡ F¯0 (·) as ε → 0.

(2.18)

Proof The following relation takes place, in the case where the condition P1 (c) is satisfied, for t ≥ 0:

2.1 RP with regenerative lifetimes

31

F¯ε (t) − Fε (t) = P{ζε,1 ≤ t} − P{ζε,1 ≤ t, με ≥ ζε,1 } = P{ζε,1 ≤ t, με ≤ ζε,1 } ≤ P{με ≤ ζε,1 } = qε → 0 as ε → 0.

(2.19) 

This relation completes the proof. Let us introduce Laplace transforms, for ε ∈ (0, 1]: ∫ ∞ ¯ φε (s) = e−su F¯ε (du), s ≥ 0,

(2.20)

0

∫

and φε (s) =

∞

0

e−su Fε (du), s ≥ 0.

(2.21)

The condition P1 can be re-formulated in the following equivalent form: ∫∞ (a) φ¯ε (s) → φ¯0 (·) as ε → 0, for s ≥ 0, where φ¯0 (s) = 0 e−su F¯0 (du) is the Laplace transform of the distribution function F¯0 (u) concentrated on the interval [0, ∞), (b) F¯0 (u) is a weakly non-arithmetic distribution function, and (c) qε → q0 = 0 as ε → 0.

P◦1 :

It is useful noting that, under the condition P◦1 (c), the convergence relation given in the condition P◦1 (a) is equivalent to the following convergence relation: φε (s) → φ0 (s) = φ¯0 (s) as ε → 0, for s ≥ 0. Let us also introduce expectations, for ε ∈ (0, 1]: ∫ ∞ e¯ε = s F¯ε (ds)

(2.22)

(2.23)

0

∫

and eε =

0

∞

sFε (ds).

(2.24)

We also assume that the following condition holds: ∫∞ Q1 : (a) e¯ε < ∞, for ε ∈ (0, 1], (b) e¯ε → e¯0 = 0 s F¯0 (ds) < ∞ as ε → 0. The following useful lemma takes place. Lemma 2.2 The conditions P1 (a), (b) and Q1 imply that eε < ∞, for ε ∈ (0, 1], and ∫ ∞ sF0 (ds) = e¯0 < ∞ as ε → 0. (2.25) eε → e0 = 0

Proof The following relation takes place, for ε ∈ (0, 1] and T ≥ 0:

32

2 Ergodic theorems for perturbed RP

0 ≤ e¯ε − eε = Eζε,1 − Eζε,1 I(με ≥ ζε,1 ) = Eζε,1 I(με < ζε,1 ) = Eζε,1 I(ζε,1 < T)I(με < ζε,1 ) + Eζε,1 I(ζε,1 ≥ T)I(με < ζε,1 ).

(2.26)

The condition P1 (c) implies that the bounded random variables ζε,1 I(ζε,1 < P

T)I(με < ζε,1 ) −→ 0 as ε → 0 and, thus, Eζε,1 I(ζε,1 < T)I(με < ζε,1 ) → 0 as ε → 0, for T ≥ 0. Taking into account this fact, the relation (2.26), conditions P1 , Q1 , and Lemma B.2, we get the following relation, for any 0 ≤ Tk → ∞ as k → ∞, which are of points of continuity for the distribution function F¯0 (·): lim (e¯ε − eε ) ≤ lim Eζε,1 I(ζε,1 ≥ Tk )I(με < ζε,1 )

ε→0

ε→0

≤ lim Eζε,1 I(ζε,1 ≥ Tk ) ε→0 ∫ ∫ ∞ s F¯ε (ds) = lim (e¯ε − = lim ε→0 Tk

∫

= e¯0 −

ε→0

Tk

Tk

0

s F¯0 (ds) → 0 as k → ∞.

s F¯ε (ds)) (2.27)

0

The relation (2.27) completes the proof.



Remark 2.1 It is useful to note that in the case, where (a) the condition P1 holds, (b) eε < ∞, for ε ∈ (0, 1], and (c) the relation (2.25) holds, the condition Q1 may not hold. Additional assumptions that (d) eε,− = Eζε,1 I(με < ζε,1 ) < ∞, for ε ∈ (0, 1] and (e) eε,− → 0 as ε → 0 are needed to the condition Q1 would be fulfilled. Let us now formulate the perturbation condition for the free term qε (t, A) of the renewal equation (2.13). Definition 2.1 P[BZ ] is a class of functions q(t, A), t ∈ R+, A ∈ BZ such that: (a) q(t, A) is a non-negative Borel measurable function of argument t ∈ R+ , for A ∈ BZ , and (b) q(t, A) is a finite measure as a function of A ∈ BZ , for t ∈ R+ . Definition 2.2 Function q(t, A), t ∈ R+, A ∈ BZ from the class P[BZ ] is majorised by the tail probability function 1 − F(t), t ∈ R+ of the distribution function concentrated on R+ if (2.28) q(t, Z) ≤ 1 − F(t), for t ∈ R+ . Definition 2.3 Function q(t, A), t ∈ R+, A ∈ BZ from the class P[BZ ] is consistent with the tail probability function 1 − F(t), t ∈ R+ of the distribution function concentrated on R+ if q(t, Z) = 1 − F(t), for t ∈ R+ . (2.29) According to the relation (2.8), the function qε (t, A) belongs to the class P[BZ ], for every ε ∈ (0, 1].

2.1 RP with regenerative lifetimes

33

Also, the function qε (t, A) is, for every ε ∈ (0, 1], majorised by the tail probability function 1 − F¯ε (·), i.e., qε (t, Z) ≤ P{ζε,1 > t} = 1 − F¯ε (t), for t ∈ R+ .

(2.30)

Let us also introduce a modified simpler variant of the function qε (t, A), for ε ∈ (0, 1]: (2.31) q¯ε (t, A) = P{ξε (t) ∈ A, ζε,1 > t}, t ∈ R+, A ∈ BZ . The function q¯ε (t, A), t ∈ R+, A ∈ BZ belongs to the class P[BZ ], for every ε ∈ (0, 1]. Also, the function q¯ε (t, A) is, for every ε ∈ (0, 1], consistent with the tail probability function 1 − F¯ε (·), i.e., qε (t, Z) = P{ζε,1 > t} = 1 − F¯ε (t), for t ∈ R+ .

(2.32)

We also assume that the following perturbation condition holds: R1 : There exists a function q¯0 (t, A), t ≥ 0, A ∈ BZ , which belongs to the class P[BZ ], a class of set Γ ⊆ BZ , and Borel sets U[q¯ · (·, A)], A ∈ Γ such that: (a) the function q¯0 (t, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function us 1 − F¯0 (t), t ∈ R+ ; (b) the functions q¯ε (·, A) −→ q¯0 (·, A) as ε → 0, for points s ∈ U[q¯ · (·, A)], A ∈ Γ; (c) m(U¯ (q¯ · (·, A))) = 0, for A ∈ Γ; (d) the function q¯0 (·, A) is continuous almost everywhere with respect to the Lebesgue measure m(·) on B+ , for A ∈ Γ. It is also helpful to comment on the above perturbation condition. It is worth noting that the notation U[q¯ · (·, A)] is used to indicate that this set of convergence is actually determined by the family of functions q¯ε (·, A), ε ∈ (0, 1]. In light of the relation (2.32), the consistency condition R1 (a) is natural. Lemma 2.3 Let the condition P1 (c) be satisfied. Then, the condition R1 is satisfied for the functions q¯ε (t, A), t ∈ R+, A ∈ B, ε ∈ (0, 1] if and only if this condition is satisfied for the functions qε (t, A), t ∈ R+, A ∈ BZ, ε ∈ (0, 1], with the limiting function q¯0 (t, A) = q0 (t, A), t ∈ R+, A ∈ BZ , the Borel sets U[q¯ · (·, A)] = U[q · (·, A)], and the class of set Γ ⊆ BZ , appearing in the condition R1 . Proof The following relation takes place, for ε ∈ (0, 1] and t ∈ R+, A ∈ B: qε (t, A) = P{ξε (t) ∈ A, ζε,1 > t, με > t} = P{ξε (t) ∈ A, ζε,1 > t} − P{ξε (t) ∈ A, ζε,1 > t, με ≤ t} = q¯ε (t, A) − qε,− (t, A),

(2.33)

where qε,− (t, A) = P{ξε (t) ∈ A, ζε,1 > t, με ≤ t}.

(2.34)

Obviously, the random event {ξε (t) ∈ A, ζε,1 > t, με ≤ t} ⊆ {με < ζε.1 } for any t ∈ R+ . This fact and the condition P1 (c) imply that the following relation takes place:

34

2 Ergodic theorems for perturbed RP

sup qε,− (t, A) ≤ P{με < ζε.1 } = qε → 0 as ε → 0. t ≥0

(2.35)

The relation (2.35) implies that the following relation of asymptotic uniform convergence takes place, for any s ∈ R+ and A ∈ BZ : us

qε,− (·, A) −→ 0(·) as ε → 0,

(2.36)

where 0(s) = 0, s ∈ R+ . It obviously follows from the relations (2.33) and (2.36) that the relation of asymptotic uniform convergence given in the condition R1 (b) holds for the functions q¯ε (t, A), t ∈ R+, A ∈ B, ε ∈ (0, 1] if and only if this asymptotic relation holds for the functions qε (t, A), t ∈ R+, A ∈ B, ε ∈ (0, 1]. Moreover, the limiting function q¯0 (t, A) = q0 (t, A), t ∈ R+, A ∈ BZ , and the sets U[q¯ · (·, A)] = U[q · (·, A)], A ∈ Γ are  the same for the functions q¯ε (t, A) and qε (t, A). The majorisation relation (2.32) implies that, for every ε ∈ (0, 1] and A ∈ BZ , the function qε (·, A) is majorised by the tail probability function 1 − F¯ε (·) on the interval R+ , i.e., (2.37) qε (t, A) ≤ 1 − F¯ε (t), for t ∈ R+ . Similarly, the consistency condition R1 (a) implies that the majorisation relation similar to (2.37) holds, for A ∈ BZ , q0 (t, A) ≤ 1 − F¯0 (t) = 1 − F0 (t), for t ∈ R+ .

(2.38)

The following useful lemma takes place. Lemma 2.4 Let the conditions P1 –R1 be satisfied. Then, the conditions Y5 , Y6 are satisfied for the distribution functions Fε (·) and the conditions Y9 , Y10 hold for the functions qε (·, A), for any A ∈ Γ. Proof The condition P1 and Lemma 2.1 imply that the condition Y5 holds for the distribution functions Fε (·). The condition P1 , Q1 and Lemma 2.2 imply that the condition Y6 holds for the distribution functions Fε (·). Let us show that, for A ∈ Γ, the conditions Y9 and Y10 hold for the functions qε (·, A), if to choose the corresponding majorising functions qε,+ (t) = 1− F¯ε (t), t ≥ 0, for ε ∈ [0, 1]. The majorisation relations (2.37) and (2.38) imply that the majorisation relations appearing in the conditions Y9 and Y10 hold. Also, according to the relation (2.37) 1 − F¯ε (t) ≤ 1, for t ≥ 0 and ε ∈ (0, 1]. Thus, the conditions Y9 (a1) and Y10 (a1) are satisfied. The conditions P1 , R1 , Lemma 2.2, and the majorisation relations (2.37) and (2.38) imply that the following relation holds:

2.1 RP with regenerative lifetimes

∫ lim

35

∞

(1 − F¯ε (s))ds ∫ T = lim lim (e¯ε − (1 − F¯ε (s))ds) 0≤T →∞ 0≤ε→0 0 ∫ T = lim (e¯0 − (1 − F¯0 (s))ds) 0≤T →∞ ∫ ∞ 0 = lim (1 − F¯0 (s))ds = 0. lim

0≤T →∞ 0≤ε→0 T

0≤T →∞ T

(2.39)

Note that symbol 0 ≤ ε → 0, which admits the case ε ≡ 0, is used in relation (2.39). The relation (2.39) implies that the conditions Y9 (a2) and Y10 (a2) hold. The condition R1 (b) coincides with the asymptotic relations of locally uniform convergence given in the condition Y10 .  The conditions R1 (c), (d) coincide with the conditions Y10 (b), (c). 2.1.2.2 The Limiting Renewal Equation. According to the condition R1 , the function q0 (t, A) belongs to class L. Also, according to the condition P1 , the distribution function F0 (·) is proper, and it is not concentrated at zero. Thus, the following standard renewal equation can be written, for every A ∈ BZ , ∫ t p0 (t, A) = q0 (t, A) + p0 (t − s, A)F0 (ds), t ≥ 0. (2.40) 0

It has the unique solution p0 (t, A), t ≥ 0, which belongs to the class L. Moreover, p0 (t, A) ∈ [0, 1], for t ≥ 0. The conditions P1 –R1 imply that the distribution functions Fε (s), their first moments eε , and the functions qε (t) converge in some sense to the corresponding limiting distribution function F0 (s), its first moment e0 , and the function q0 (t) as ε → 0, respectively. This allows us to consider the equation (2.6) for ε ∈ (0, 1] as a perturbed version of the equation (2.40) and interpret ε as a perturbation parameter. 2.1.2.3 Structure of the Class Γ. The condition R1 implies that the class Γ can always be extended to its maximal form such that Γ would be closed with respect to the operations of the union of disjoint sets, the difference of sets connected by the relation of inclusion, and the complement. Let Γ ⊆ BZ be the maximal class of sets A ∈ BZ , for which the condition R1 holds. Lemma 2.5 Let the conditions P1 and R1 be satisfied. Then, the maximal class Γ: (a) contains the phase space Z and (b) is closed with respect to the operations of union for disjoint sets, difference for sets connected by the relation of inclusion, and complement. Proof The class Γ appearing in condition R1 includes the phase space Z.

36

2 Ergodic theorems for perturbed RP

Indeed, q¯ε (t, Z) = P{ζε,1 > t} = 1 − F¯ε (t), t ∈ R+ , according to the relation (2.32). The function 1 − F¯ε (t), t ∈ R+ is monotonic, for every ε ∈ [0, 1], and thus, the relation of weak convergence given in condition P1 (a) implies that, for s ∈ C[F0 (·)], us q¯ε (·, Z) = 1 − F¯ε (·) −→ 1 − F¯0 (·) = q¯0 (·, Z) as ε → 0.

(2.41)

¯ 0 (·)] = [0, ∞) \ C[F0 (·)] is at most a countable set, m(C[F ¯ 0 (·)] = 0. Since, C[F Thus, the conditions R1 (b) and (c) hold for the functions q¯ε (t, Z). Since, q0 (t, Z) = 1 − F0 (t), t ∈ R+ , the condition R1 (d) also holds for the function q¯0 (·, Z). Thus, the space Z belongs to the class Γ. The condition R1 implies that the class Γ appearing in the condition R1 is closed with respect to the operation of union for disjoint sets, i.e., if the condition R1 holds for sets A and A such that A ∩ A = ∅, then this condition also holds for the set A = A ∪ A. Indeed, qε (t, ·), t ∈ R+ is, for every ε ∈ [0, 1] a finite measure, q¯ε (t, A ∪ A) = q¯ε (t, A) + q¯ε (t, A), for any t ∈ R+ . Let U[q¯ · (·, A)] and U[q¯ · (·, A)] be the corresponding sets of convergence appearing in the condition R1 (b). Obviously the following relation takes place, for s ∈ U[q¯ · (·, A)] ∩ U[q¯ · (·, A)] and any 0 ≤ sε → s as ε → 0: q¯ε (sε, A ∪ A) = q¯ε (sε, A) + q¯ε (sε, A)

→ q¯0 (s, A) + q¯0 (s, A) = q¯0 (s, A ∪ A) as ε → 0.

(2.42)

Therefore, the conditions R1 (b) and (c) hold for the functions q¯ε (·, A ∪ A), with the set of convergence U[q¯ · (·, A ∪ A)] = U[q¯ · (·, A)] ∩ U[q¯ · (·, A)]. According to the condition R1 (b), m(U¯ [q¯ · (·, A)]) = m(U¯ [q¯ · (·, A)]) = 0, and, thus, m(U[q¯ · (·, A ∪ A)]) = m(U¯ [q¯ · (·, A)] ∪ (U¯ [q¯ · (·, A)]) = 0. Also, the condition R1 (d) holds for the function q¯0 (·, A ∪ A) = q¯0 (·, A) + q¯0 (·, A), since this condition holds for the functions q¯0 (·, A) and q¯0 (·, A). Similarly, one can prove that the class Γ appearing in condition R1 is closed with respect to the operation of differences for sets connected by relation of inclusion, i.e., if the condition R1 holds for sets A and A such that A ⊆ A, then this condition also holds for the set A = A \ A. Finally, by choosing A = A, A = Z, one can prove that the class Γ is also closed with respect to the complement operation, i.e., if the condition R1 holds for a set A, then it also holds for the set A¯ = Z \ A.  Let us, for example, consider the model, where the phase space Z = {1, 2, . . . , M } is a finite set and BZ is the σ-algebra of all subsets of Z. In this case, it is natural to assume that all one-point sets, A = { j}, j ∈ Z, belong to the class Γ. This obviously implies that any subset A ⊆ Z belongs to the class Γ, that is, Γ = BZ . 2.1.2.4 The Case of Existence of the Limiting Regenerative Process. It is worth noting that the conditions O1 –R1 do not imply the existence of a limiting regenerative process.

2.1 RP with regenerative lifetimes

37

However, suppose that there exist independent triplets ξ¯0,n = ξ0,n (t), t ≥ 0 , κ0,n, μ0,n , n = 1, 2, . . ., defined on some probability space Ω0, F0, P0 and satisfying the model assumptions (A) - (E). Also, let the regenerative process ξ0 (t), t ≥ 0, with regeneration times ζ0,n, n = 0, 1, . . ., and regenerative lifetime μ0 be constructed using these triplets and relations (2.1)–(2.4). It can be assumed that the corresponding limiting characteristics in the conditions P1 –R1 are defined via the regenerative process ξ0 (t), t ≥ 0. In this case, the distribution function F¯0 (·) = P{κ0,1 ≤ ·} is not concentrated at zero, and the stopping probability q0 = P{μ0,1 < κ0,1 } = 0. P Since F¯0 (0) < 1, the random variables ζ0,n −→ ∞ as n → ∞, and, thus, the regenerative process ξ0 (t) is well-defined on the time interval [0, ∞). Let us introduce probabilities: P0 (t, A) = P{ξ0 (t) ∈ A}, A ∈ BZ, t ≥ 0.

(2.43)

Obviously, P0 (t, A) ∈ [0, 1], A ∈ BZ, t ≥ 0. Also, the function P0 (t, A), t ≥ 0, belongs, for every A ∈ BZ , to the class L, and it is unique in the class L solution for the following renewal equation: ∫ t P0 (t, A) = q0 (t, A) + P0 (t − s, A)F0 (ds), t ≥ 0, (2.44) 0

where, for t ≥ 0, F0 (t) = F¯0 (t) = P{ζ0,1 ≤ t} = P{κ0,1 ≤ t},

(2.45)

and, for A ∈ BZ, t ≥ 0, q0 (t, A) = P{ξ0 (t) ∈ A, ζ0,1 > t} = P{ξ0,1 (t) ∈ A, κ0,1 > t}.

(2.46)

Note also that the function q0 (t, A), t ≥ 0, belongs, for every A ∈ BZ , to the class L and q0 (t, A) ∈ [0, 1], A ∈ BZ, t ≥ 0. It can be assumed that the corresponding limiting characteristics in the conditions P1 –R1 age given by relations (2.46) and (2.45). Relation (2.46) obviously implies that, in this case, the function q0 (t, A) belongs to the class P[BZ ] and is consistent with the distribution function F0 (·). Thus, the condition R1 (a) holds for the function q0 (t, A). In this case, the condition R1 (a) can be omitted in the condition R1 . Since, the stopping probability q0 = 1 − F0 (∞) = 0, the equation (2.44) is the standard renewal equation. In fact, this equation coincides with the renewal equation (2.40), and, thus, the solutions of these two renewal equations P0 (t, A) = p0 (t, A), t ≥ 0, for A ∈ BZ .

38

2 Ergodic theorems for perturbed RP

2.2 Ergodic Theorems for Perturbed Regenerative Processes with Regenerative Lifetimes In this section, we define the corresponding limiting stationary distributions and present ergodic theorems for perturbed regenerative processes with regenerative lifetimes.

2.2.1 Ergodic Theorems for Perturbed Regenerative Processes 2.2.1.1 Ergodic Theorems for Perturbed Regenerative Processes with Regenerative Lifetimes. According to Lemma 12.4, the conditions P1 –R1 imply that, for every A ∈ Γ, the conditions of Theorem A.1 hold for the renewal equation (2.40) and, thus, the following relation takes place: ∫ ∞ 1 q0 (s, A)ds as t → ∞. (2.47) p0 (t, A) → π0 (A) = e0 0 Let us define function π0 (A), for A ∈ BZ : ∫ ∞ 1 π0 (A) = q0 (s, A)m(ds). e0 0

(2.48)

Note that the Lebesgue integration is used in expression on right hand side of relation (2.48) instead of the direct Riemann integration used in the expression on the right hand side of the relation (2.47). The finite Lebesgue integral on the right hand side of the relation (2.48) exists since, by the conditions P1 –R1 , the function q0 (·, A) belongs to the class L and is majorised, for every A ∈ BZ , by the tail probability function 1 − F0 (·), for which ∫∞ (1 − F (s))m(ds) = e0 < ∞, 0 0 Moreover, π0 (A), A ∈ BZ , is a probability measure on σ-algebra BZ , since, according to the condition R1 , the function q0 (s, A), s ∈ R+, A ∈ BZ belongs to the class P and is consistent with the tail probability function 1 − F0 (t), t ≥ 0. This measure is a limiting stationary distribution for the perturbed regenerative processes ξε (t). According to the condition R1 , the function q0 (·, A) is directly Riemann integrable, for every A ∈ Γ, and, thus, the Lebesgue integration in (2.48) can be replaced by the direct Riemann integration, as it is done in the relation (2.47), that is, for A ∈ Γ, ∫ ∞ ∫ ∞ 1 1 q0 (s, A)m(ds) = q0 (s, A)ds. (2.49) π0 (A) = e0 0 e0 0 From the above remarks, it follows that the limits π0 (A), A ∈ Γ defined by the elation (2.47) for the maximal class Γ (see, Sect. 2.1.2.3) have the following properties: (a) π0 (Z) = 1, (b) π0 (A ∪ A) = π0 (A) + π0 (A), for A, A ∈ Γ, A ∩

2.2 Ergodic theorems for perturbed RP with regenerative lifetimes

39

A = ∅, (c) π0 (A \ A) = π0 (A) − π0 (A), for A, A ∈ Γ, A ⊆ A, and (d) ¯ = 1 − π0 (A), A ∈ Γ. π0 ( A) Moreover, if the class Γ is a σ-algebra, then π0 (A), A ∈ Γ defined by the relation (2.47) is a probability measure on Γ. The conditions O1 –R1 imply that, for every A ∈ Γ, the conditions of Theorem A.3 hold for the perturbed renewal equation (2.6). Application of Theorem A.3 to this equation gives the following ergodic theorem for perturbed regenerative processes with regenerative lifetimes. Theorem 2.1 Let the conditions O1 –R1 be satisfied. Then, for every A ∈ Γ, and any 0 ≤ tε → ∞ as ε → 0 such that qε tε → t ∈ [0, ∞] as ε → 0, Pε (tε, A) → e−t/e0 π0 (A) as ε → 0.

(2.50)

2.2.1.2 Ergodic Theorems for Regenerative Processes with Regenerative Lifetimes and Transition Periods. Theorem 2.1 can be generalised to the model of perturbed regenerative processes with transition periods. Let us additionally assume that the following condition holds: P 1 : (a) Fε (·) ⇒ F0 (·) as ε → 0, where F0 (·) is a proper distribution function. Let us define the Laplace transform, for ε ∈ [0, 1], ∫ ∞ φε (s) = e−st Fε (dt), s ≥ 0.

(2.51)

0

It is useful noting that the following condition is equivalent to the condition P¯ 1 : P ◦1 : (a) φε (s) → φ0 (s) as ε → 0, for s ≥ 0, where φ˜0 (s) is the Laplace transform of a proper distribution function F0 (·). Let us also define the stopping probability for the transition period, for ε ∈ [0, 1]: qε = P{με,1 < ζε,1 } = 1 − Fε (∞).

(2.52)

The condition P 1 obviously implies that the stopping probabilities for transition period: (2.53) qε → q0 = 0 as ε → 0. It is also useful noting that, for A ∈ BZ and T ≥ 0, sup qε (s, A) ≤ P{ζε,1 ∧ με,1 > T }

s ≥T

= P{ζε,1 > T, με,1 ≥ ζε,1 } + P{ζε,1 ∧ με,1 > F, με,1 < ζε,1 } ≤ P{με,1 ≥ ζε,1 } − P{ζε,1 ≤ T, με,1 ≥ ζε,1 } + P{με,1 < ζε,1 } = Fε (∞) − Fε (T) + qε . (2.54) From the relations (2.53), (2.54) and the condition P 1 , it follows that

40

2 Ergodic theorems for perturbed RP

lim lim sup qε (s, A) = 0.

T →0 ε→0 s ≥T

(2.55)

The following ergodic theorem for perturbed regenerative processes with regenerative lifetimes and transition periods is the direct corollary of the renewal Theorem A.5 for the perturbed renewal equation. Theorem 2.2 Let the conditions O1 –R1 and P 1 be satisfied. Then, for every A ∈ Γ, and any 0 ≤ tε → ∞ as ε → 0 such that qε tε → t ∈ [0, ∞] as ε → 0, Pε (tε, A) → e−t/e0 π0 (A) as ε → 0.

(2.56)

In the case of standard regenerative processes, Theorem 2.2 just reduces to Theorem 2.1. Indeed, the condition P 1 can be omitted since it is implied by the condition P1 . The ergodic relation (2.56) reduces to the ergodic relation (2.50).

2.2.2 Ergodic Theorems for Perturbed Regenerative Processes with Modified Regenerative Lifetimes 2.2.2.1 Modified Regenerative Lifetimes. Let us also introduce modified regenerative lifetimes: ν

νε ε −1

κε,k , με,+ = κε,k (2.57) με,− = k=1

k=1

and consider probabilities: Pε,± (t, A) = P{ξε (t) ∈ A, με,± > t}, A ∈ BZ, t ≥ 0.

(2.58)

με,− ≤ με ≤ με,+,

(2.59)

Pε,− (t, A) ≤ Pε (t, A) ≤ Pε,+ (t, A).

(2.60)

Obviously, and, thus, for any A ∈ BZ, t ≥ 0,

The following theorem is a useful modification of Theorem 2.2. Theorem 2.3 Let the conditions O1 –R1 and P 1 be satisfied. Then, for every A ∈ Γ, and any 0 ≤ tε → ∞ as ε → 0 such that qε tε → t ∈ [0, ∞] as ε → 0, Pε,± (tε, A) → e−t/e0 π0 (A) as ε → 0.

(2.61)

Proof First of all note that in the case where qε = 0, the random variables με,− = με = με,+ = ∞ with probability 1, and therefore, the probabilities Pε,− (t, A) = Pε (t, A) = Pε,+ (t, A), for any A ∈ BZ, t ≥ 0. Thus, for any sequence 0 < εn → 0 as n → ∞ such that qεn = 0, n = 1, 2, . . ., the relation (2.61) follows from the corresponding version of the relation (2.56) given in Theorem 2.2.

2.2 Ergodic theorems for perturbed RP with regenerative lifetimes

41

From the above remark, let us assume that 0 < qε → 0 for ε ∈ (0, 1]. It is obvious that the random variable νε is geometrically distributed with parameter qε , i.e., P{νε = n} = (1 − qε )n qε, n = 1, 2, . . .. It is well-known that, in this case, the following asymptotic relation holds: d

qε νε −→ ν0 as ε → 0, where ν0 is an exponentially distributed random variable with parameter 1. This relation obviously implies that the following relation holds: d (2.62) (qε (νε − 1), qε νε ) −→ (ν0, ν0 ) as ε → 0. The conditions P1 and Q1 imply that for any 0 < ne → ∞ as ε → 0, the following  d relation holds: n1ε k ≤tnε κε,k , t ≥ 0 −→ e0 t, t ≥ 0, as ε → 0. A similar statement (with a proof) is given in relation (9.55)1 . This relation obviously imply that the following relation holds: (

1

1

d κε,k , κε,k ), t ≥ 0 −→ (e0 t, e0 t), t ≥ 0 as ε → 0. nε k ≤tn nε k ≤tn ε

(2.63)

ε

The relations (2.62) and (2.63) and Theorem B.4 imply that the following relation holds: 1

1

(qε (νε − 1), qε νε, κε,k , κε,k ), t ≥ 0 nε k ≤tn nε k ≤tn ε

ε

d

−→ (ν0, ν0, e0 t, e0 t), t ≥ 0 as ε → 0.

(2.64)

The relations (2.62), (2.63), where one should choose nε = qε−1 , and Theorem B.1 imply that the following relation holds:



(qε με,−, qε με,+ ) = (qε κε,k , qε κε,k ) k ≤νε −1

k ≤νε

d

−→ (e0 ν0, e0 ν0 ) as ε → 0.

(2.65) −1





The bivariate distribution function P{ν0 ≤ s , ν0 ≤ s  } = 1 − e−e0 (s ∧s ), s , s  ≥ 0, is continuous. Taking this fact into account, we get, using relation (2.65), the following relation, for A ∈ BZ and t = limε→0 qε tε ∈ [0, ∞): Pε,+ (tε, A) − Pε,− (tε, A) = P{ξε (tε ) ∈ A, με,+ > tε } − P{ξε (t) ∈ A, με,− > tε } = P{ξε (t) ∈ A, με,+ > tε, με,− ≤ tε } ≤ P{qε με,+ > qε tε, qε με,− ≤ qε tε } = P{qε με,− ≤ qε tε } − P{qε με,+ ≤ qε tε, με,− ≤ qε te } −1

−1

→ (1 − e−e0 t ) − (1 − e−e0 t ) = 0 as ε → 0.

(2.66)

42

2 Ergodic theorems for perturbed RP

The relation (2.65) also implies that, for any 0 ≤ sn → ∞ as n → ∞ and A ∈ BZ and t = limε→0 qε tε = ∞, lim Pε,+ (tε, A) ≤ lim P{με,+ > tε }

ε→0

ε→0

= lim P{qε με,+ > qε tε } ε→0

≤ lim P{qε με,+ > sn } ε→0

−1 s

= e−e0

n

→ 0 as n → ∞.

(2.67)

The relation (2.56) given in Theorem 2.2 and the relations (2.60), (2.66), and (2.67) imply, in an obvious way, that the relation (2.61) holds.  2.2.2.2 Quasi-Ergodic Theorems for Regenerative Processes. Theorems 2.1– 2.3 can be alternatively referred as quasi-ergodic theorems for regenerative processes. In particular, Theorems 2.1 and 2.2 are modifications of the quasi-ergodic theorems for perturbed regenerative processes presented in Gyllenberg and Silvestrov (2000a, 2008). The main difference is that the existence of limiting regenerative processes is not assumed in Theorems 2.1–2.3. This also leads to modification of the corresponding perturbation conditions and proofs. Theorem 2.3 is new. We would like to mention the important case, where stopping probability qε = 0, ε ∈ (0, 1]. In this case, the regenerative stopping time με = ∞ with probability 1. Also, qε tε → 0 as ε → 0, for any 0 ≤ tε → ∞ as ε → 0. In this case, the probability Pε (t, A) = P{ξε (t) ∈ A} is a one-dimensional distribution for the process ξε (t). Theorems 2.1 and 2.2 are, in this case, usual individual ergodic theorem for perturbed regenerative processes ξε (t). It is also worth to mention the case of an unperturbed regenerative process. This is the case, where ξε (t) = ξ0 (t), t ≥ 0, for ε ∈ (0, 1]. The conditions O1 –R1 reduce in this case to the minimal conditions of the standard individual ergodic theorem for regenerative processes, which directly follows from the classical renewal theorem given in Feller (1971): ∫ ∞ (a) F0 (·) is a weakly non-arithmetic distribution function; (b) expectation e0 = 0 sF0 (ds) < ∞; and (c) the function q0 (s, A), s ≥ 0, is, for A ∈ Γ, continuous almost everywhere with respect to the Lebesgue measure m(ds) on B+ . Note that q0 (s, A) ≤ 1 − F0 (s), s ≥ 0, and, thus condition (c) is, under the above condition (b), equivalent to the assumption of the direct Riemann integrability for the function q0 (s, A), s ≥ 0. Also, the condition P 1 just reduces to the assumption that (d) F0 (·) is a proper distribution function. The corresponding individual ergodic theorem takes in this case the form of the asymptotic relation (2.47), that is, for A ∈ Γ, P0 (t, A) → π0 (A) as t → ∞.

(2.68)

Chapter 3

Perturbed Alternating Regenerative Processes

In this chapter, we introduce the model of perturbed alternating regenerative processes modulated by regularly or singularly perturbed two-state semi-Markov processes. The chapter includes three sections. In Sect. 3.1, we introduce the model of perturbed alternating regenerative processes modulating by perturbed two-states semi-Markov processes and formulate the corresponding basic perturbation conditions. In Sect. 3.2, we formulate conditions separating regularly, singularly, and supersingularly perturbed alternating regenerative processes. We also comment on and clarify the meaning of super-long, long, and short time ergodic theorems for singularly and super-singularly perturbed alternating regenerative processes, which describe the ergodic behaviour of distributions for such processes in various asymptotic time zones. In Sect. 3.3, we describe important procedures of time compression and aggregation of regeneration times for perturbed alternating regenerative processes.

3.1 Alternating Regenerative Processes In this section, we introduce the model of perturbed alternating regenerative processes and formulate the corresponding basic perturbation conditions used in ergodic theorems for such processes.

3.1.1 Alternating Regenerative Processes As in Sect. 2.1.1, let Ωε, Fε, Pε be, for every ε ∈ (0, 1], a probability space. We assume that all stochastic processes and random variables introduced below and indexed by parameter ε are defined on the probability space Ωε, Fε, Pε . Let us assume the following model assumptions are fulfilled: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0_3

43

44

3 Perturbed ARP

(F) ξ¯ε,i,n = ξε,i,n (t), t ≥ 0 be, for every i = 1, 2 and n = 1, 2, . . ., a measurable stochastic process with a phase space Z. (G) κε,i,n be, for every i = 1, 2 and n = 1, 2, . . ., a non-negative random variable. (H) ηε,i,n and ηε be binary random variables taking values in the space X = {1, 2}, for every i = 1, 2 and n = 1, 2, . . .. (I) Stochastic triplets ξ¯ε,i,n = ξε,i,n (t), t ≥ 0 , κε,i,n, ηε,i,n , i = 1, 2, n = 1, 2, . . . and the random variable ηε are mutually independent. (J) Joint distributions of random variables ξε,i,n (tk ), k = 1, . . . , r, and κε,i,n, ηε,i,n do not depend on n ≥ 1, for every i = 1, 2 and tk ∈ [0, ∞), k = 1, . . . , r, r ≥ 1. Here, the assumption about measurability of the processes ξ¯ε,i,n is absolutely similar to those formulated in the model assumption (A) for the processes ξ¯ε,n , in Sect. 2.1.1.1. Let us define recurrently stochastic sequences of switching binary random indices: ηε,n = ηε,ηε, n−1,n, n = 1, 2, . . . , ηε,0 = ηε,

(3.1)

ζε,n = κε,ηε,0,1 + · · · + κε,ηε, n−1,n, n = 1, 2, . . . , ζε,0 = 0,

(3.2)

regeneration times:

and modulated alternating regenerative process, with two components: ξε (t) = ξε,ηε, n−1,n (t − ζε,n−1 ) and ηε (t) = ηε,n−1, for t ∈ [ζε,n−1, ζε,n ), n = 1, 2, . . . .

(3.3)

The above model assumptions (F)–(J) imply that the modulating index sequence ηε,n, n = 0, 1, . . ., is a homogeneous Markov chain with the phase space X = {1, 2}, an initial distribution p¯ε = pε,i = P{ηε,0 = i}, i = 1, 2 , and transition probabilities: pε,i j = P{ηε,1 = j/ηε,0 = i} = P{ηε,i,1 = j}, i, j = 1, 2.

(3.4)

The above model assumptions (F)–(J) also imply that the modulating index process ηε (t), t ≥ 0, is a semi-Markov process with the phase space X and transition probabilities: Q ε,i j (t) = Pi {ζε,1 ≤ t, ηε,1 = j} = P{κε,i,1 ≤ t, ηε,i,1 = j}, t ≥ 0, i, j = 1, 2.

(3.5)

Let us also introduce the distribution functions of sojourn times for semi-Markov process ηε (t), for i ∈ X: Fε,i (t) = Pi {ζε,1 ≤ t} = P{κε,i,1 ≤ t}, = Q ε,i1 (t) + Q ε,i2 (t), t ≥ 0.

(3.6)

3.1 ARP

45

We assume that the following condition holds: O2 : (a) pε,i j = 0, ε ∈ (0, 1] or pε,i j > 0, ε ∈ (0, 1], for i, j ∈ X and (b) Fε,i (0) = 0 for every ε ∈ (0, 1] and i ∈ X. The condition O2 (b) obviously implies that, for every ε ∈ (0, 1], the random P

variables ζε,n −→ ∞ as n → ∞ and, thus, the alternating regenerative process (ξε (t), ηε (t)) is well-defined on the time interval [0, ∞). The object of our interest is joint distributions: Pε,i j (t, A) = Pi {ξε (t) ∈ A, ηε (t) = j}, A ∈ BZ, i, j ∈ X, t ≥ 0.

(3.7)

The probabilities Pε,i j (t, A), i ∈ X, for every A ∈ BZ, j ∈ X, are measurable functions of t ≥ 0, which are the unique bounded solution for the following system of renewal type equations: Pε,i j (t, A) = I(i = j)qε,i (t, A) 2 ∫ t

+ Pε,k j (t − s, A)Q ε,ik (ds), t ≥ 0, i ∈ X, k=1

(3.8)

0

where, for A ∈ BZ, t ≥ 0, i, j ∈ X, qε,i (t, A) = Pi {ξε (t) ∈ A, ζε,1 > t} = P{ξε,i,1 (t) ∈ A, κε,i,1 > t}.

(3.9)

Note also that qε,i (t, A) ∈ [0, 1], A ∈ BZ, t ≥ 0, and function qε,i (t, A), t ≥ 0, belongs, for every A ∈ BZ and i ∈ X, to the class L.

3.1.2 Perturbation Conditions for Alternating Regenerative Processes 3.1.2.1 Perturbation Conditions for Alternating Regenerative Processes. Let us introduce, for i ∈ X and ε ∈ (0, 1], sets: Yε,i = { j ∈ X : pε,i j > 0}.

(3.10)

It is useful to note that the condition O2 (a) implies that, for i ∈ X and ε ∈ (0, 1], Yε,i = Y1,i .

(3.11)

Also, let us introduce conditional distribution functions, defined for j ∈ Yε,i, i ∈ X and ε ∈ (0, 1]: Q ε,i j (t)/pε,i j for t ≥ 0, j ∈ Y1,i, i ∈ X, Fε,i j (t) = (3.12) ¯ 1,i, i ∈ X. for t ≥ 0, j ∈ Y Fε,i (t)

46

3 Perturbed ARP

Note that the condition O2 implies that Fε,i j (0) = 0, i, j ∈ X, for ε ∈ (0, 1]. We also assume that the following perturbation condition holds: P2 : (a) pε,i j → p0,i j as ε → 0, for i, j ∈ X, (b) Q ε,i j (·) ⇒ Q0,i j (·) as ε → 0, for i, j ∈ X, where Q0,i j (·) is a proper or improper distribution function such that Q0,i j (∞) = p0,i j , for i, j ∈ X, and (c) F0,i j (·) = p−1 0,i j Q 0,i j (·) is a non-arithmetic distribution function, for i, j ∈ X such that p0,i j > 0. The asymptotic relation Fε (·) ⇒ F0 (·) as ε → 0 (applied to proper or improper distribution functions Fε (·)) means that Fε (t) → F0 (t) as ε → 0, for any t ∈ R1 , which is a point of continuity for the limiting distribution function F0 (·). Since the 2 × 2 matrix Pε = pε,i j  is stochastic, the condition P2 (a) implies that the matrix P0 = p0,i j  is also stochastic. Let η0,n, n = 0, 1, . . ., be a Markov chain with the phase space X and the matrix of transition probabilities P0 . The condition P2 (a) makes it possible to interpret the Markov chains ηε,n , for ε ∈ (0, 1], as a perturbed version of the Markov chain η0,n . Also, the conditions P2 (a) and (b) imply that the limiting transition probabilities Q0,i j (t), t ≥ 0, i, j ∈ X, can serve as transition probabilities of some two-state semiMarkov process η0 (t), t ≥ 0. The conditions P2 (a) and (b) make it possible to interpret the semi-Markov process ηε (t) as a perturbed version of the semi-Markov process η0 (t), for ε ∈ (0, 1]. Note also that the condition P2 (a) and the relation (3.11) imply that, for i ∈ X, Y0,i = { j ∈ X : p0,i j > 0} ⊆ Y1,i .

(3.13)

The conditions P2 (a) and (b) imply that, for j ∈ Y0,i, i ∈ X, Fε,i j (·) ⇒ F0,i j (·) as ε → 0, and, thus, for i ∈ X,



Fε,i (·) =

Q ε,i j (·) +

j ∈Y0, i

⇒



(3.14)

Q ε,i j (·)

¯ 0, i j ∈Y



Q0,i j (·) = F0,i (·) as ε → 0.

(3.15)

j ∈Y0, i

Obviously, for i ∈ X, F0,i (·) =

j ∈Y0, i

Q0,i j (·) =



F0,i j (·)p0,i j .

(3.16)

j ∈Y0, i

The relation (3.16) and the condition P2 (c) obviously imply that the distribution function F0,i (·) is non-arithmetic, for i ∈ X. ¯ 0,i, i ∈ X, Also, the condition P2 (a) implies that, for t ≥ 0 and j ∈ Y Q ε,i j (t) ≤ Q ε,i j (∞) = pε,i j → p0,i j = 0 as ε → 0.

(3.17)

3.1 ARP

47

¯ 0,i, i ∈ Thus, the asymptotic relation given in the condition P2 (b) holds for j ∈ Y X, with the limiting probabilities Q0,i j (t) = 0, t ≥ 0, i.e., for t ≥ 0, Q ε,i j (t) → Q0,i j (t) ≡ 0 as ε → 0.

(3.18)

The following useful lemma explains the relationship between the condition P2 and the conditions G, I, and J formulated in Sect. 9.1.21 . Lemma 3.1 Let the conditions G, I, and J (with the normalisation functions u ·,i ≡ 1, i ∈ X) are satisfied. Then: (i) The condition P2 (b) holds with the limiting transition probabilities: F0,i j (·)p0,i j for j ∈ Y1,i, i ∈ X, (3.19) Q0,i j (·) = ¯ 1,i, i ∈ X. 0(·) ≡ 0 for j ∈ Y (ii) p−1 0,i j Q 0,i j (·) = F0,i j (·), for i, j ∈ X such that p0,i j > 0. (iii) If, additionally, F0,i j (·) is a non-arithmetic distribution function, for i, j ∈ X such that p0,i j > 0, then the condition P2 is satisfied. Proof The conditions I and P2 (a) coincide. The conditions I and J imply that, for j ∈ Y1,i, i ∈ X, Q ε,i j (·) = Fε,i j (·)pε,i j ⇒ F0,i j (·)p0,i j = Q0,i j (·) as ε → 0.

(3.20)

¯ 1,i, i ∈ X, Also, the condition G implies that, for j ∈ Y Q ε,i j (t) ≤ Q ε,i j (∞) = pε,i j ≡ 0,

(3.21)

Q ε,i j (·) ⇒ Q0,i j (·) = 0(·) ≡ 0 as ε → 0.

(3.22)

and, thus, Note that, for i, j ∈ X, Q0,i j (t) → Q0,i j (∞) = p0,i j as t → ∞.

(3.23)

Thus, the condition P2 (b) holds. Since the set Y0,i ⊆ Y1,i , for i ∈ X, the proposition (ii) holds due to the relation (3.19). The above remarks, obviously, imply that the proposition (iii) also holds.  The condition P2 (b) can be reformulated in terms of Laplace transforms, defined for i, j ∈ X and ε ∈ (0, 1], ∫ ∞ e−su Q ε,i j (du), s ≥ 0. (3.24) ψε,i j (s) = 0

Let also define conditional Laplace transforms, for i, j ∈ X and ε ∈ (0, 1]:

48

3 Perturbed ARP

∫ φε,i j (s) =

0

∫

and ψε,i (s) =

∞

0

∞

e−su Fε,i j (du), s ≥ 0,

(3.25)

e−su Fε,i (du), s ≥ 0.

(3.26)

According to the relation (3.12), φε,i j (s)pε,i j for s ≥ 0, j ∈ Y1,i, i ∈ X, ψε,i j (s) = ¯ 1,i, i ∈ X. for s ≥ 0, j ∈ Y ψε,i (s)

(3.27)

The condition P2 is equivalent to the following condition: P◦2 : (a) pε,i j → p0,i j as ε → 0, for i, j = 1, 2, (b) ψε,i j (s) → ψ0,i j (s) as ε → 0, for s ≥ ∫∞ 0 and i, j ∈ X, where ψ0,i j (s) = 0 e−su Q0,i j (du), s ≥ 0 is the Laplace transform of some some proper or improper distribution such that Q0,i j (∞) = p0,i j , for ∫∞ −1 −su i, j ∈ X, (c) φ0,i j (s) = p0,i j ψ0,i j (s) = 0 e F0,i j (du), s ≥ 0 is the Laplace transform of a non-arithmetic distribution function F0,i j (·) = p−1 0,i j Q 0,i j (·), for i, j ∈ X such that p0,i j > 0. It worth noting that the condition P◦2 (a), in fact, is implied by the condition P◦2 (b), since φε,i j (0) = pε,i j , i, j ∈ X, for ε ∈ [0, 1]. However, in view of the importance of the convergence relations for the transition probabilities pε,i j , it makes sense to include these relations in the condition P◦2 . Let us introduce expectations, for i, j ∈ X and ε ∈ (0, 1]: eε,i j = Ei ζε,1 I(ηε,1 = j) = E κε,i,1 I(ηε,i,1 = j) =

∫ 0

∞

sQ ε,i j (ds), i, j ∈ X.

(3.28)

Hereinafter, we use the notations Pi and Ei for conditional probabilities and expectations under the condition ηε (0) = ηε = i. We also impose the following condition of convergence for the above expectations: ∫∞ Q2 : (a) eε,i j < ∞, for ε ∈ (0, 1] and i, j ∈ X; and (b) eε,i j → e0,i j = 0 sQ0,i j (ds) < ∞ as ε → 0, for i, j ∈ X. The conditions Q2 and P2 (c) imply that, for j ∈ Y0,i, i ∈ X, e0,i j ∈ (0, ∞).

(3.29)

¯ 0,i, i ∈ X, Also, the condition Q2 and the relation (3.18) imply that, for j ∈ Y e0,i j = 0.

(3.30)

We also introduce the expectation of the sojourn time for the semi-Markov process ηε (t), for i ∈ X and ε ∈ (0, 1]:

3.1 ARP

49

eε,i = Ei ζε,1 = E κε,i,1 =



eε,i j .

(3.31)

The condition Q2 obviously implies that, for i ∈ X,

eε,i → e0,i = e0,i j as ε → 0.

(3.32)

j ∈X

j ∈Y0, i

The relations (3.29) and (3.32) imply that, for i ∈ X, e0,i ∈ (0, ∞).

(3.33)

The following useful lemma explains the relationship between the condition Q2 and the conditions G, I, J, and K formulated in Sect. 9.1.21 . Lemma 3.2 Let the conditions G, I, J, and K (with the normalisation functions u ·,i ≡ 1, i ∈ X) be satisfied. Then the condition Q2 is satisfied, with the limits of expectations, ∫ ∞ f0,i j p0,i j for j ∈ Y1,i, i ∈ X, sQ0,i j (ds) = e0,i j = (3.34) ¯ 1,i, i ∈ X. 0 for j ∈ Y 0 Proof The conditions I, J, and K imply that the following relation holds, for j ∈ Y1,i, i ∈ X: (3.35) eε,i j = fε,i j pε,i j → f0,i j p0,i j = e0,i j as ε → 0. Please note that in this case, e0,i j = f0,i j p0,i j ∫ ∞ ∫ = sF0,i j (ds)p0,i j = 0

0

∞

sQ0,i j (ds).

(3.36)

¯ 1,i, i ∈ X, and, thus, by relation Also, Q ε,i j (∞) = pε,i j = 0, ε ∈ (0, 1], for j ∈ Y ¯ 1,i, i ∈ X: (3.28), the following relation holds, for j ∈ Y eε,i j = 0 = e0,i j , for ε ∈ (0, 1]. ∫∞ Indeed, in this case, Q0,i j (·) ≡ 0(·) and, thus, e0,i j = 0 sQ0,i j (ds) = 0.

(3.37) 

According to the relation (3.9), the functions qε,i (t, A), t ∈ R+, A ∈ BZ belong to the class P[BZ ], for i ∈ X, ε ∈ (0, 1]. Moreover, the function qε,i (t, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − Fε,i (t), for i ∈ X and ε ∈ (0, 1], i.e., qε,i (t, Z) = 1 − Fε,i (t), for t ∈ R+ . We also assume that the following perturbation condition is satisfied:

(3.38)

50

3 Perturbed ARP

R2 : There exist functions q0,i (t, A), t ≥ 0, A ∈ BZ , for i ∈ X, which belong to class P[BZ ], a class of set Γ ⊆ BZ , and Borel sets U[q ·,i (·, A)], A ∈ Γ, i ∈ X such that: (a) function q0,i (t, A), t ∈ R+, A ∈ BZ is consistent with the tail probability us function 1 − F0,i (·), for i ∈ X; (b) functions qε,i (·, A) −→ q0,i (·, A) as ε → 0, for points s ∈ U[q ·,i (·, A)], A ∈ Γ, i ∈ X; (c) m(U¯ (q ·,i (·, A))) = 0, for A ∈ Γ, i ∈ X; and (d) function q0,i (·, A) is continuous almost everywhere with respect to the Lebesgue measure m(·) on B+ , for A ∈ Γ, i ∈ X. It is worth to note that notation U[q ·,i (·, A)] is used in order to indicate that this set of convergence is, actually, determined by the family of functions qε,i (·, A), ε ∈ (0, 1]. Also, it is useful to make some comments concerning the above perturbation condition. In the light of relations (3.38), the consistency condition R2 (a) is natural. The consistency relation (2.32) implies that, for every ε ∈ (0, 1] and A ∈ BZ, i ∈ X, function qε,i (·, A) is majorised by the tail probability function 1−Fε,i (·) on interval [0, ∞), i.e., (3.39) qε,i (t, A) ≤ 1 − Fε,i (t), for t ∈ R+ . The condition R2 (a) implies that the following majorisation relation holds, similar to (3.39), for A ∈ BZ, i ∈ X: q0,i (t, A) ≤ 1 − F0,i (t), for t ∈ R+ .

(3.40)

The following useful lemma takes place. Lemma 3.3 Let the conditions P2 –R2 be satisfied. Then, for every A ∈ Γ, i ∈ X, the conditions Y5 , Y6 and Y9 , Y10 are satisfied for the distribution functions Fε,i (·) and the functions qε,i (·, A). 3.1.2.2 Structure of the Class Γ. The condition R2 implies that the class Γ can always be extended to its maximal form such that Γ would be closed with respect to the operations of the union of disjoint sets, the difference of sets connected by the relation of inclusion, and the complement. Let Γ ⊆ BZ be the maximal class of sets A ∈ BZ , for which the condition R2 is satisfied. Lemma 3.4 Let the conditions P2 and R2 be satisfied. Then, the maximal class Γ: (a) contains the phase space Z and (b) is closed with respect to the operations of the union of disjoint sets, the difference for sets connected by the relation of inclusion, and the complement. Proof The proof of this lemma is similar to the proof of Lemma 2.2. The class Γ appearing in condition R2 includes the phase space Z. Indeed, for ε ∈ (0, 1] and i ∈ X, (3.41) qε,i (t, Z) = P{ζε,i,1 > t} = 1 − Fε,i (t), for t ∈ R+ . The condition P2 implies that, for any tε → t as ε → 0, where t ∈ C[F0,i (·)] (here, C[F0,i (·)] is the set of continuity points for the distribution function F0,i (·)), and i ∈ X,

3.1 ARP

51

1 − Fε,i (tε ) → 1 − F0,i (t) as ε → 0.

(3.42)

The relations (3.41) and (3.42) imply that, for any 0 ≤ sε → s ∈ C[F0,i (·)] as ε → 0, and i ∈ X, qε,i (·, Z) = 1 − Fε,i (·) us

−→ 1 − F0,i (·) = q0,i (·, Z) as ε → 0.

(3.43)

¯ 0,i (·)], i ∈ X, are at most countable sets, m(C[F ¯ 0,i (·)]) = 0. Since C[F Thus, the conditions R2 (b) and (c) hold for the functions qε,i (·, Z), i ∈ X, with the sets U[q0,i (·, Z)] = C[F0,i (·)], i ∈ X. Since, q0,i (t, Z) = 1 − F0,i (t), t ∈ R+ , for i ∈ X, the condition R2 (d) also holds for the functions q0,i (·, Z), i ∈ X. Thus, the space Z belongs to the class Γ. From the condition R2 , it follows that the class Γ appearing in the condition R2 is closed with respect to the union operation for disjoint sets, that is, if the condition R2 holds for the sets A and A such that A ∩ A = ∅, then this condition also holds for the set A = A ∪ A. Indeed, qε,i (t, ·), t ∈ R+ , is, for every i ∈ X and ε ∈ [0, 1], a finite measure, and, thus, qε,i (t, A ∪ A) = qε,i (t, A) + qε,i (t, A) for any t ∈ R+ . Let U[q ·,i (·, A)] and U[q ·,i (·, A)] be the corresponding sets of convergence appearing in condition R2 (b). Obviously the following relation takes place for s ∈ U[q ·,i (·, A)] ∩ U[q ·,i (·, A)] and 0 ≤ sε → s as ε → 0, i ∈ X: qε,i (sε, A ∪ A) = qε,i (sε, A) + qε,i (sε, A) → q0,i (s, A) + q0,i (s, A) = q0,i (s, A ∪ A) as ε → 0.

(3.44)

According to the condition R2 (c), m(U¯ [q ·,i (·, A)]) = m(U¯ [q ·,i (·, A)]) = 0, i ∈ X, and, thus, m(U¯ [q ·,i (·, A)] ∪ (U¯ [q ·,i (·, A)]) = 0. Therefore, the conditions R2 (b) and (c) hold for functions qε,i (·, A ∪ A), i ∈ X, with the corresponding sets of convergence, U[q ·,i (·, A ∪ A)] = U[q ·,i (·, A)] ∩ U[q0,i (·, A )], i ∈ X. Also, the condition R2 (d) holds for the functions q0,i (·, A ∪ A) = q0,i (·, A) + q0,i (·, A), i ∈ X, since this condition holds for the functions q0,i (·, A), q0,i (·, A), i ∈ X. Similarly, one can prove that the class Γ appearing in the condition R2 is closed with respect to the difference operation for sets connected by the inclusion relation, that is, if the condition R2 holds for the sets A and A such that A ⊆ A, then this condition holds for the set A = A \ A. Finally, by choosing A = A, A = Z, one can prove that the class Γ is also closed with respect to the complement operation, that is, if the condition R2 holds for set A, then it also holds for the set A¯ = Z \ A.  3.1.2.3 The Case of Existence of the Limiting Alternating Regenerative Process. It is worth noting that the conditions O2 –R2 do not imply the existence of a limiting regenerative process.

52

3 Perturbed ARP

However, suppose that there exist independent triplets ξ¯0,i,n = ξ0,i,n (t), t ≥ 0 , κ0,i,n, η0,i,n , n = 1, 2, . . . , i ∈ X, and a random variable η0 defined on some probability space Ω0, F0, P0 and satisfying the model assumptions (F)–(J). Let, also, an alternating regenerative process (ξ0 (t), η0 (t)), t ≥ 0 with the regeneration times ζ0,n, n = 0, 1, . . ., and a modulating semi-Markov component η0 (t), t ≥ 0 be constructed using these triplets and relations (3.1)–(3.3). It can be assumed that the corresponding limiting characteristics in the conditions P2 –R2 are defined via the alternating regenerative processes (ξ0 (t), η0 (t)), t ≥ 0. In this case, the distribution functions F0,i (·) = P{κ0,i,1 ≤ ·}, i ∈ X, are not concentrated at zero. P Thus, the random variables ζ0,n −→ ∞ as n → ∞, and, therefore, the alternating regenerative process (ξ0 (t), η0 (t)) is well-defined on the time interval [0, ∞). Let us introduce probabilities: P0,i j (t, A) = Pi {ξ0 (t) ∈ A, η0 (t) = j}, A ∈ BZ, i, j ∈ X, t ≥ 0.

(3.45)

The probabilities P0,i j (·, A), i ∈ X, belong, for every A ∈ BZ, j ∈ X, to the class L and are the unique in this class solution for the following system of renewal type equations: P0,i j (t, A) = I(i = j)q0,i (t, A) 2 ∫ t

+ P0,k j (t − s, A)Q0,ik (ds), t ≥ 0, i ∈ X, k=1

(3.46)

0

where, for t ≥ 0, i, j ∈ X, Q0,i j (t) = Pi {ζ0,1 ≤ t, η0,1 = j} = P{κ0,i,1 ≤ t, κ0,i,1 = j}, t ≥ 0,

(3.47)

and, for A ∈ BZ, t ≥ 0, i, j ∈ X, q0,i (t, A) = Pi {ξ0 (t) ∈ A, ζ0,1 > t} = P{ξ0,i,1 (t) ∈ A, κ0,i,1 > t}.

(3.48)

Note that q0,i (t, A) ∈ [0, 1], A ∈ BZ, t ≥ 0, and the function q0,i (t, A), t ≥ 0, belongs to the class L, for every A ∈ BZ and i ∈ X. In this case, it should be assumed that the corresponding limiting characteristics in the conditions P2 –R2 age given by the relations (3.48) and (3.47). The relation (3.48) implies that the function q0,i (t, A) belongs to the class P[BZ ]  and is consistent with the distribution function F0,i (·) = j ∈X Q0,i j (·), for i ∈ X. Thus, the condition R2 (a) holds for the functions q0,i (t, A), t ∈ R+, A ∈ BZ , for i ∈ X. In this case, the condition R2 (a) can be omitted in the condition R2 . 3.1.2.4 Ergodic Theorems for Alternating Regenerative Processes with Degenerated Switching Random Variables. Let us consider the case, where the

3.1 ARP

53

switching random variables ηε,i,n = i, n ≥ 1 with probability 1, for some i ∈ X and ε ∈ (0, 1]. Let us also consider, for every ε ∈ (0, 1], i ∈ X, a standard regenerative process ξε,i (t), t ≥ 0 with regeneration times: ζε,i,n = κε,i,1 + · · · + κε,i,n, n = 1, 2, . . . , ζε,i,0 = 0,

(3.49)

defined by the following recurrent relations: ξε,i (t) = ξε,i,n (t − ζε,i,n−1 ), for t ∈ [ζε,i,n−1, ζε,in ), n = 1, 2, . . . .

(3.50)

The condition O2 , implies that, for ε ∈ (0, 1], i = 1, 2, the random variables P

ζε,i,n −→ ∞ as ε → 0 and, thus, the process ξε,i (t) is well-defined on the time interval [0, ∞). The probabilities pε,i (t, A) = P{ξε,i (t) ∈ A} ∈ [0, 1], t ≥ 0 are, for ε ∈ (0, 1], A ∈ BZ, i ∈ X, the unique in the class L solution of the renewal equation: ∫ t pε,i (t, A) = qε,i (t, A) + pε,i (t − s, A)Fε,i (ds), t ≥ 0. (3.51) 0

According to the condition R2 , the function q0,i (t, A) belongs to the class P[Z], for every i ∈ X. According to the condition P2 , F0,i (·), i ∈ X are proper distribution functions, which are not concentrated at zero. Thus, the following standard renewal equation can be written, for every A ∈ Γ and i ∈ X: ∫ t p0,i (t, A) = q0,i (t, A) + p0,i (t − s, A)F0,i (ds), t ≥ 0. (3.52) 0

This equation has the unique solution p0,i (t, A), t ≥ 0, that belongs to the class L. Moreover, p0,i (t, A) ∈ [0, 1], t ≥ 0. The conditions P1 –R1 imply that the distribution functions Fε (s), their first moments eε , and the functions qε (t) converge in some sense to the corresponding limiting distribution function F0 (s), its first moment e0 , and the function q0 (t) as ε → 0, respectively. This allows us to consider the equation (2.6) for ε ∈ (0, 1] as a perturbed version of the equation (2.40) and interpret ε as a perturbation parameter. The conditions O2 –R2 imply that the functions q0,i (·, A), A ∈ Γ, i ∈ X, belong to the class L and are directly Riemann integrable. Also, the first moments e0,i < ∞, i ∈ X. Therefore, for every A ∈ Γ, i ∈ X, the conditions of Theorem A.2 hold for the renewal equation (3.51). From Theorem A.1 it follows that the following asymptotic relation holds, for every A ∈ Γ, i ∈ X and 0 ≤ tε → ∞ as ε → 0: ∫ ∞ 1 pε,i (tε, A) → π0,i (A) = q0,i (s, A)ds as t → ∞. (3.53) e0,i 0

54

3 Perturbed ARP

The stationary distributions π0,i (·), i = 1, 2 are used in formulas for the stationary distributions in ergodic theorems for perturbed alternating regenerative processes presented in Chaps. 4–9. Let us define the function π0,i (A), A ∈ BZ , for i ∈ X: ∫ ∞ 1 π0,i (A) = q0,i (s, A)m(ds). (3.54) e0,i 0 Note that the Lebesgue integration is used in the expression on right hand side of the relation (3.54) instead of the direct Riemann integration used in the expression on right hand side of the relation (3.53). The finite Lebesgue integral on the right hand side of relation (3.54) exists since, by the conditions P2 –R2 , the functions q0,i (·, A), i ∈ X, belong to the class L and are majorised, for every A ∈ BZ , by the tail probability function 1 − F0,i (·), for which ∫∞ (1 − F (s))m(ds) = e0,i < ∞, i ∈ X. 0,i 0 Moreover, π0,i (A), A ∈ BZ , is, for i ∈ X, a probability measure on the σ-algebra BZ , since, according to the condition R1 , the function q0,i (s, A), s ∈ R+, A ∈ BZ belongs to the class P[Z] and is consistent with the tail probability function 1 − F0,i (s), s ≥ 0, for i ∈ X. This measure is the limiting stationary distribution for the perturbed regenerative processes ξε,i (t), for i ∈ X. According to the condition R2 , the functions q0,i (·, A) are also directly Riemann integrable for A ∈ Γ, and, thus, the Lebesgue integration in (3.54) can be replaced for A ∈ Γ by the direct Riemann integration, as it is done in the relation (3.53), that is, for A ∈ Γ, i ∈ X, ∫ ∞ ∫ ∞ 1 1 q0,i (s, A)m(ds) = q0,i (s, A)ds. (3.55) π0,i (A) = e0,i 0 e0,i 0 From the above remarks, it follows that the limits π0,i (A), A ∈ Γ, defined by the relation (2.47) for the maximal class Γ (see, Sect. 3.1.2.2) have the following properties, for i ∈ X: (a) π0 (Z) = 1, (b) π0 (A ∪ A) = π0 (A) + π0 (A), for A ∩ A = ∅, A, A ∈ Γ, (c) π0,i (A \ A) = π0,i (A) − π0,i (A), for A ⊆ A = ¯ = 1 − π0,i (A), A ∈ Γ. ∅, A, A ∈ Γ, and (d) π0,i ( A) Moreover, if the class Γ is a σ-algebra, then function π0,i (A), A ∈ Γ, defined by relation (3.53) is a probability measure on Γ, for i ∈ X.

3.2 Regularly, Singularly, and Super-Singularly Perturbed Alternating Regenerative Processes In this section we present the models of regularly, singularly, and super-singularly perturbed alternating regenerative processes and comment possible types of the socalled super-long, long, and short time ergodic theorems for perturbed alternating regenerative processes.

3.2 Regularly, singularly, and super-singularly perturbed ARP

55

3.2.1 Regular, Singular, and Super-Singular Perturbation Models for Alternating Regenerative Processes Our goal is to conduct a detailed analysis of individual ergodic theorems for the probabilities Pε,i j (t, A), that is, to describe possible options for their asymptotic behaviour as t → ∞ and ε → 0. We shall see that the asymptotic behaviour of the transition probabilities pε,i j for the Markov chains ηε,n plays an important role in these ergodic theorems. According to the condition P2 (a), these transition probabilities converge to the corresponding limiting transition probabilities p0,i j of some Markov chain η0,n , as ε → 0, that is, for i, j ∈ X, (3.56) pε,i j → p0,i j as ε → 0. Also, the following additional balancing condition, which should be assumed to hold for some β ∈ [0, ∞], play a key role in the following asymptotic analysis: S β : βε =

pε,12 pε,21

→ β as ε → 0.

There are three classes of perturbed alternating regenerative processes with significantly different ergodic properties. The first class includes the so-called regularly perturbed alternating regenerative processes, for which the limiting Markov chain η0,n is ergodic. In this case, it is equivalent to the assumption that at least one of its transition probabilities p0,12 and p0,21 is positive. According to the condition P2 (a), three cases are possible: (a) p0,12, p0,21 > 0. (b) p0,12 = 0, p0,21 > 0. (c) p0,12 > 0, p0,21 = 0. In these cases, condition Sβ obviously holds, and, moreover, the parameter β takes the following form: p0,12 . (3.57) β= p0,21 In the case (a), β ∈ (0, ∞), and, thus, the phase space X is one class of communicative states. In this case, the corresponding stationary probabilities: α1 (β) =

p0,21 p0,12 1 1 , α2 (β) = = = . p0,12 + p0,21 1 + β p0,12 + p0,21 1 + β−1

(3.58)

In the case (b), β = 0, and, thus, the phase space X consists of the absorbing state 1 and the transient state 2. In this case, α1 (0) = 1, α2 (0) = 0.

(3.59)

Similarly, in the case (c), one should count β = ∞, the phase space X consists of the absorbing state 2 and the transient state 1. In this case,

56

3 Perturbed ARP

α1 (∞) = 0, α2 (∞) = 1.

(3.60)

In the cases (b) and (c) we can also classify the corresponding alternating regenerative processes as semi-regularly perturbed and consider the class of these processes as a sub-class of regularly perturbed alternating regenerative processes. In ergodic theorems for perturbed alternating regenerative processes, the asymptotic stability of stationary probabilities for Markov chains ηε,n plays a key role. In the case of regularly perturbed models, the condition P2 (a) obviously implies that the Markov chain ηε,n is ergodic, for every ε ∈ [0, 1]. Its stationary probabilities are determined by parameter: βε =

pε,12 . pε,21

(3.61)

Namely, α1 (βε ) =

pε,21 pε,12 1 1 = , α2 (βε ) = = . pε,12 + pε,21 1 + βε pε,12 + pε,21 1 + βε−1

(3.62)

The condition P2 (a) implies that

and, thus, for i = 1, 2,

βε → β as ε → 0,

(3.63)

αi (βε ) → αi (β) as ε → 0.

(3.64)

We shall see that individual ergodic theorems for regularly perturbed alternating processes have the form of the following asymptotic relation, which holds for A ∈ Γ, i, j ∈ X and any 0 ≤ tε → ∞ as ε → 0: (β)

Pε,i j (tε, A) → π0, j (A) as ε → 0. (β)

(3.65)

The limiting probabilities π0, j (A) depend on parameter β ∈ [0, ∞], but they do not depend on the initial state i ∈ X. The forms of ergodic theorems are similar to those, which are known for unperturbed alternating regenerative processes. The second and the third classes include the so-called singularly and supersingularly perturbed alternating regenerative processes, for which the limiting Markov chain η0,n is not ergodic. It is equivalent to the assumption that both transition probabilities p0,12 and p0,21 equal 0. According to the condition P2 (a), four cases are possible: (d) 0 < pε,12, pε,21 → 0 as ε → 0. (e) pε,12 = 0, ε ∈ [0, 1] and 0 < pε,21 → 0 as ε → 0. (f) 0 < pε,12 → 0 as ε → 0 and pε,21 = 0, ε ∈ [0, 1]. (g) pε,12, pε,21 = 0, ε ∈ [0, 1]. The case (d) corresponds to singularly perturbed alternating regenerative processes.

3.2 Regularly, singularly, and super-singularly perturbed ARP

57

Three cases (e), (f), and (g) correspond to super-singularly perturbed alternating regenerative processes. In the case (d), the asymptotic stability for the stationary probabilities α j (βε ), j = 1, 2, is provided by the following condition: S β : βε =

pε,12 pε,21

→ β ∈ [0, ∞] as ε → 0.

The condition P2 (a) implies that the Markov chain ηε,n is ergodic, for ε ∈ (0, 1]. Its stationary probabilities are determined by the parameter βε , namely, α1 (βε ) =

pε,21 pε,12 1 1 = , α2 (βε ) = = . pε,12 + pε,21 1 + βε pε,12 + pε,21 1 + βε−1

(3.66)

The conditions P2 (a) and Sβ imply that, for i = 1, 2, αi (βε ) → αi (β) as ε → 0, where α1 (β) =

1 1 , α2 (β) = . 1+β 1 + β−1

(3.67)

(3.68)

In the case (e), βε = pε,12 /pε,21 = 0, for ε ∈ (0, 1], and, thus, the condition S0 holds. In this case, the Markov chain ηε,n is ergodic, for ε ∈ (0, 1] and its stationary probabilities: (3.69) αε,1 (0) = 1, αε,2 (0) = 0, for ε ∈ (0, 1]. Also, the following relation obviously holds, for i = 1, 2: αε,i (0) → αi (0) as ε → 0,

(3.70)

α1 (0) = 1, α2 (0) = 0.

(3.71)

where Similarly, in the case (f), βε = pε,12 /pε,21 = ∞, for ε ∈ (0, 1], and, thus, the condition S∞ holds. In this case, the Markov chain ηε,n is ergodic, for ε ∈ (0, 1], and its stationary probabilities: (3.72) αε,1 (0) = 0, αε,2 (0) = 1, for ε ∈ (0, 1]. Also, the following relation obviously holds, for i = 1, 2: αε,i (∞) → αi (∞) as ε → 0,

(3.73)

α1 (∞) = 0, α2 (∞) = 1.

(3.74)

where

58

3 Perturbed ARP

3.2.2 Super-Long, Long, and Short Time Ergodic Theorems for Perturbed Alternating Regenerative Processes Ergodic theorems for singularly or super-singularly perturbed alternating regenerative processes have much more complex and interesting forms than for regularly perturbed alternating regenerative processes. The following functions play important roles of the so-called time compression factors, respectively, for singularly and super-singularly perturbed models: −1 −1 vε = p−1 ε,12 + pε,21, ε ∈ (0, 1] and wε = (pε,12 + pε,21 ) , ε ∈ (0, 1].

(3.75)

In the case (d), 0 < wε < vε < ∞, for ε ∈ (0, 1] and wε, vε → ∞ as ε → 0. In the cases (e) and (f), 0 < wε < vε = ∞, for ε ∈ (0, 1] and wε → ∞ as ε → 0. The main individual ergodic theorems for singularly perturbed alternating regenerative processes take the form of the following asymptotic relation, which (under the assumption that the condition Sβ holds for some β ∈ [0, ∞]) holds, for A ∈ Γ, i, j = 1, 2, and any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → t ∈ [0, ∞] as ε → 0: (β) (3.76) Pε,i j (tε, A) → π0,i j (t, A) as ε → 0. The asymptotic behaviour of the probabilities Pε,i j (tε, A) can differ for different asymptotic time zones determined by the asymptotic relation tε /vε → t ∈ [0, ∞]. (β) The corresponding limiting probabilities π0,i j (t, A) may depend on t ∈ [0, ∞], the parameter β ∈ [0, ∞], appearing in condition Sβ , and, also, on the initial state i ∈ X, if t ∈ [0, ∞). It is natural to classify the corresponding theorems as super-long, long,, and short time ergodic theorems, for the cases t = ∞, t ∈ (0, ∞), and t = 0, respectively. In these cases, the limiting stationary probabilities take different analytical forms. The corresponding individual ergodic theorems for super-singularly perturbed alternating regenerative processes take the form of the similar asymptotic relation, which (under the assumption that the condition S0 or S∞ holds and, thus, β = 0 or β = ∞) holds, for A ∈ Γ, i, j = 1, 2 and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → t ∈ [0, ∞] as ε → 0: (β)

Pε,i j (tε, A) → π0,i j (t, A) as ε → 0.

(3.77)

In this case, the asymptotic behaviour of the probabilities Pε,i j (tε, A) also can differ for different asymptotic time zones determined by the asymptotic relation (β) tε /wε → t ∈ [0, ∞]. The corresponding limiting probabilities π0,i j (t, A) may depend on t ∈ [0, ∞], the parameter β (taking, as it was mentioned above, one of two values 0 or ∞), and, also, on the initial state i ∈ X, if t ∈ [0, ∞). As for singularly perturbed models, it is natural to classify the corresponding theorems as super-long, long, and short time ergodic theorems, for the cases t = ∞, t ∈ (0, ∞), and t = 0, respectively. In these cases the corresponding limiting probabilities take different analytical forms.

3.3 Time compression and aggregation of regeneration times

59

Ergodic theorems for singularly perturbed models, for the cases where the condition S0 or S∞ is satisfied, can be compared with ergodic theorems for super-singularly perturbed models, respectively, for the cases (e) or (f). Indeed, as was mentioned above, the condition S0 or S∞ is satisfied in the case (e) or (f), respectively. In the cases (e) and (f), i.e., for super-singularly perturbed models, vε = ∞, while 0 < wε < ∞, for ε ∈ (0, 1]. The only factor wε can be used as a time scaling factor. In the case (d), i.e., for singularly perturbed models, 0 < wε < vε < ∞, for ε ∈ (0, 1]. The question arises if wε can be used as a time scaling factor instead of vε . The answer is yes, if the condition Sβ holds for some β ∈ (0, ∞). Indeed, in this case, wε /vε → β(1 + β)−2 ∈ (0, ∞) as ε → 0. The asymptotic relations, tε /vε → t as ε → 0 and tε /wε → t as ε → 0, generate, in some sense, equivalent asymptotic time zones. However, the answer for the above question is no, if the condition S0 or S∞ is satisfied. Indeed, in this case, wε /vε → 0 as ε → 0. The asymptotic relations, tε /vε → t as ε → 0 and tε /wε → t as ε → 0, generate different asymptotic time zones, in the corresponding ergodic theorems. This, actually, makes it possible to obtain (under the assumption that the condition S0 or S∞ is satisfied) additional ergodic relations for singularly perturbed processes, similar to those given above for super-singularly perturbed processes, for the asymptotic time zones generated by the relation tε /wε → t as ε → 0. The extremal case (g) corresponds to absolutely singular perturbed alternating regenerative processes. This case is not covered by the condition Sβ . However, in this case, the modulating process ηε (t) = ηε (0), t ≥ 0. Respectively, the process ξε (t), t ≥ 0, coincides with the standard regenerative process ξε,i (t), t ≥ 0, if ηε (0) = i. The corresponding ergodic theorem for process ξε,i (t) is given by Theorem 2.1, for its particular case described in Sect. 2.2.1.3. It takes the form of the following asymptotic relation holding for A ∈ Γ, i = 1, 2 and any 0 ≤ tε → ∞: Pε,i (tε, A) → π0,i (A) as ε → 0.

(3.78)

3.3 Time Compression and Aggregation of Regeneration Times for Perturbed Alternating Regenerative Processes In this section, we describe the procedures for time compression and refresh time aggregation. These procedures play a key role in proving ergodic theorems for perturbed alternating regenerative processes.

60

3 Perturbed ARP

3.3.1 Time Compression for Perturbed Regenerative and Alternating Regenerative Processes 3.3.1.1 Time Compression for Perturbed Regenerative Processes. Let us return back to the model of perturbed regenerative processes with regenerative lifetimes introduced in Sect. 2.1.1. So, let ξ¯ε,n = ξε,n (t), t ≥ 0 , κε,n, με,n , n = 1, 2, . . ., be, for every ε ∈ (0, 1], stochastic triplets possessing properties (A)–(E) and ξε (t), t ≥ 0, be a regeneration process with regeneration times ζε,n, n = 0, 1, . . ., and a regenerative lifetime με constructed using the above triplets and relations (2.1)–(2.4). Let, also, uε, ε ∈ (0, 1] be a positive function. In some cases, it can be useful to replace, for every ε ∈ (0, 1], the above initial triplets by new ones ξ¯ε,uε ,n = ξε,uε ,n (t), t ≥ 0 , κε,uε ,n, με,uε ,n , n = 1, 2, . . ., where, for n = 1, 2, . . ., −1 ξε,uε ,n (t) = ξε,n (tuε ), t ≥ 0, κε,uε ,n = u−1 ε κε,n, με,uε ,n = uε με,n .

(3.79)

Using the above stochastic triplets and relations similar to the relations (2.1)–(2.4), we can define, for every ε ∈ (0, 1], a new transformed regenerative process: ξε,uε (t) = ξε,uε ,n (t − ζε,uε ,n−1 ) for t ∈ [ζε,uε ,n−1, ζε,uε ,n ), n = 1, 2, . . . ,

(3.80)

with regeneration times: ζε,uε ,n = κε,uε ,1 + · · · + κε,uε ,n, n = 1, 2, . . . , ζε,uε ,0 = 0,

(3.81)

and a regenerative lifetime: με,uε =

νε,

u ε −1

κε,uε ,k + με,uε ,νε, u ε I(νε,uε < ∞),

(3.82)

k=1

where νε,uε = min(n ≥ 1 : με,uε ,n < κε,uε ,n ).

(3.83)

The above regenerative process ξε,uε (t), t ≥ 0, can, also, be defined, for every ε ∈ (0, 1], as the transformation of the regenerative process ξε (t), t ≥ 0, namely, ξε,uε (t) = ξε (tuε ), t ≥ 0,

(3.84)

with the new transformed regenerative times: ζε,uε ,n = u−1 ε ζε,n, n = 0, 1, . . . ,

(3.85)

and the new transformed regenerative lifetime: με,uε = u−1 ε με .

(3.86)

3.3 Time compression and aggregation of regeneration times

61

It worth noting that, for ε ∈ (0, 1], the probabilities: Pε,uε (t, A) = P{ξε,uε (t) ∈ A, με,uε > t} = P{ξε (tuε ) ∈ A, με > tuε } = Pε (tuε, A), A ∈ BZ, t ≥ 0.

(3.87)

The basic renewal equation (2.6) for the probabilities Pε,uε (t, A) takes, for ε ∈ (0, 1], the following form, for A ∈ BZ : ∫ t Pε,uε (t, A) = qε,uε (t, A) + Pε,uε (t − s, A)Fε,uε (ds), t ≥ 0, (3.88) 0

where, for t ≥ 0, Fε,uε (t) = P{ζε,uε ,1 ≤ t, με,uε ≥ ζε,uε ,1 } −1 = P{ζε,1 ≤ tuε, u−1 ε με ≥ uε ζε,1 },

(3.89)

and, for A ∈ BZ, t ≥ 0, qε,uε (t, A) = P{ξε,uε (t) ∈ A, ζε,uε ,1 ∧ με,uε > t} = P{ξε,1 (tuε ) ∈ A, ζε,1 ∧ με,1 > tuε } = qε (tuε, A).

(3.90)

The time compression transformation described above can be applied in the case, where the conditions O1 –R1 do not hold for the original regenerative processes ξε (t) with regeneration times ζε,n and regenerative lifetimes με , but there exists a time compression factor uε such that the conditions O1 –R1 do hold for the compressed in time regenerative processes ξε,uε (t) with the regeneration times ζε,uε ,n and the regenerative lifetimes με,uε . In this case, Theorems 2.1–2.3 can be applied to these processes and give the corresponding ergodic relations for probabilities Pε (tuε, A). 3.3.1.2 The Case of Existence of the Limiting Regenerative Process for Compressed in Time Perturbed Regenerative Processes. This is the case, where it is assumed that there exist some limiting stochastic triplets ξ¯0,n = ξ0,n (t), t ≥ 0 , κ0,n , μ0,n , n = 1, 2, . . . possessing the properties (A)–(E) and the corresponding limiting regenerative process is defined using these stochastic triplets and relations similar to the relations (2.1)–(2.4), i.e., ξ0 (t) = ξ0,n (t − ζ0,n−1 ) for t ∈ [ζ0,n−1, ζ0,n ), n = 1, 2, . . . ,

(3.91)

ζ0,n = κ0,1 + · · · + κ0,n, n = 1, 2, . . . , ζ0,0 = 0.

(3.92)

where The assumption of holding the conditions O1 –R1 for the regenerative processes ξε,uε (t) implies that the stopping probability q0 = P{μ0,1 < ζ0,1 } = 0 and, thus, the regenerative lifetime P{μ0 = ∞} = 1. The basic renewal equation (2.6) for the probabilities P0 (t, A) = P{ξ0 (t) ∈ A} takes the following form, for A ∈ BZ :

62

3 Perturbed ARP

∫ P0 (t, A) = q0 (t, A) +

t

P0 (t − s, A)F0 (ds), t ≥ 0,

(3.93)

0

where, for t ≥ 0,

F0 (t) = P{ζ0,1 ≤ t},

(3.94)

q0 (t, A) = P{ξ0 (t) ∈ A, ζ0,1 > t}.

(3.95)

and, for A ∈ BZ, t ≥ 0,

In this case, it should be assumed that (a) characteristics of the stochastic triplets ξ¯ε,uε ,n = ξε,uε ,n (t), t ≥ 0 , κε,uε ,n, με,uε ,n , n = 1, 2, . . . replace, for every ε ∈ (0, 1], the corresponding characteristics of the original stochastic triples ξ¯ε,n = ξε,n (t), t ≥ 0 , κε,n, με,n , n = 1, 2, . . ., in the conditions O1 –R1 , and (b) the characteristics of the limiting stochastic triplets ξ¯0,n = ξ0,n (t), t ≥ 0 , κ0,n, μ0,n , n = 1, 2, . . . replace the corresponding limiting characteristics in the conditions O1 –R1 . 3.3.1.3 Time Compression for Perturbed Alternating Regenerative Processes. Let us return back to the model of perturbed alternating regenerative processes introduced in Sect. 3.1.1. So, let us assume that, for every ε ∈ (0, 1], there exist stochastic triplets ξ¯ε,i,n = ξε,i,n (t), t ≥ 0 , κε,i,n, ηε,i,n , i = 1, 2, n = 1, 2, . . . and a random variable ηε possessing the properties (F)–(J), and (ξε (t), ηε (t)), t ≥, 0 is an alternating regenerative process with regeneration times ζε,n, n = 0, 1, . . ., constructed with the use the above triplets and relations (3.1)–(3.3). Let also uε, ε ∈ (0, 1] be a positive function. In some cases, it can be useful to replace, for every ε ∈ (0, 1], the above triplets by new ones ξ¯ε,uε ,i,n = ξε,uε ,i,n (t), t ≥ 0 , κε,uε ,i,n, ηε,uε ,i,n , i = 1, 2, n = 1, 2, . . ., where, for i = 1, 2, n = 1, 2, . . ., ξε,uε ,i,n (t) = ξε,i,n (tuε ), t ≥ 0, κε,uε ,i,n = u−1 ε κε,i,n, ηε,uε ,i,n = ηε,i,n .

(3.96)

Using the above stochastic triplets and relations analogous to relations (3.1)– (3.3), we can define, for every ε ∈ (0, 1], a new transformed alternating regenerative process: (3.97) ηε,uε ,n = ηε,uε ,ηε, u ε , n−1,n, n = 1, 2, . . . , ηε,uε ,0 = ηε, with regeneration times: ζε,uε ,n = κε,uε ,ηε, u ε ,0,1 + · · · + κε,uε ,ηε, u ε , n−1,n, n = 1, 2, . . . , ζε,uε ,0 = 0,

(3.98)

and a modulated alternating regenerative process with two components: ξε,uε (t) = ξε,uε ,ηε, u ε , n−1,n (t − ζε,uε ,n−1 ) and ηε,uε (t) = ηε,uε ,n−1, for t ∈ [ζε,uε ,n−1, ζε,uε ,n ), n = 1, 2, . . . .

(3.99)

3.3 Time compression and aggregation of regeneration times

63

The above alternating regenerative process (ξε,uε (t), ηε,uε (t)), t ≥ 0, can be also defined, for every ε ∈ (0, 1], as the transformation of the alternating regenerative process (ξε (t), ηε (t)), t ≥ 0, namely, (ξε,uε (t), ηε,uε (t)) = (ξε (tuε ), ηε (tuε )), t ≥ 0,

(3.100)

with the new transformed regenerative times: ζε,uε ,n = u−1 ε ζε,n, n = 0, 1, . . . .

(3.101)

It worth noting that, for ε ∈ (0, 1], the probabilities: Pε,uε ,i j (t, A) = Pi {ξε,uε (t) ∈ A, ηε,uε (t) = j} = P{ξε (tuε ) ∈ A, ηε (tuε ) = j} = Pε,i j (tuε, A), A ∈ BZ, i, j ∈ X, t ≥ 0.

(3.102)

The basic system of renewal type equations (3.8) for the probabilities Pε,uε ,i j (t, A) takes, for ε ∈ (0, 1], the following form, for A ∈ BZ, j ∈ X: Pε,uε ,i j (t, A) = I(i = j)qε,uε ,i (t, A) 2 ∫ t

+ Pε,uε ,k j (t − s, A)Q ε,uε ,ik (ds), t ≥ 0, i ∈ X, k=1

(3.103)

0

where, for t ≥ 0, i, j ∈ X, Q ε,uε ,i j (t) = P{ζε,uε ,1 ≤ t, ηε,uε ,1 = j} = P{ζε,i,1 ≤ tuε, ηε,i,1 = j},

(3.104)

and, for A ∈ BZ, t ≥ 0, i, j ∈ X, qε,uε ,i (t, A) = Pi {ξε,uε (t) ∈ A, ηε,uε (t) = i, ζε,uε ,1 > t} = P{ξε,i,1 (tuε ) ∈ A, ζε,i,1 > tuε } = qε,i (tuε, A).

(3.105)

The time compression transformation described above can be applied in the case, where the conditions O2 –R2 do not hold for the original alternating regenerative processes (ξε (t), ηε (t)) with the regeneration times ζε,n , but there exists a time compression factor uε such that the conditions O2 –R2 do hold for the compressed in time alternating regenerative processes (ξε,uε (t), ηε,uε (t)), with the regeneration times ζε,uε ,n . In this case, the theorems given in Chaps. 4–9 can be applied to these processes and give the corresponding ergodic relations for the probabilities Pε,i j (tuε, A). 3.3.1.4 The Case of Existence of the Limiting Alternating Regenerative Process for Compressed in Time Perturbed Alternating Regenerative Processes. This is the case, where it is assumed that there exist some limiting stochastic triplets ξ¯0,i,n = ξ0,n (t), t ≥ 0 , κ0,i,n , η0,i,n , i = 1, 2, n = 1, 2, . . . and a random variable

64

3 Perturbed ARP

η0 possessing the properties (F)–(J) and the corresponding alternating limiting regenerative process is defined using of the above stochastic triplets and relations similar to the relations (3.1)–(3.3), i.e., η0,n = ηε,η0, n−1,n, n = 1, 2, . . . , η0,0 = η0,

(3.106)

the regeneration times: ζ0,n = κ0,η0,0,1 + · · · + κ0,η0, n−1,n, n = 1, 2, . . . , ζ0,0 = 0,

(3.107)

and the modulated alternating regenerative process with two components: ξ0 (t) = ξ0,η0, n−1,n (t − ζ0,n−1 ) and η0 (t) = η0,n−1, for t ∈ [ζ0,n−1, ζ0,n ), n = 1, 2, . . . .

(3.108)

The basic system of renewal type equations (3.8) for the probabilities P0,i j (t, A) =

Pi {ξ0 (t) ∈ A, η0 = j} takes the following form, for A ∈ BZ, j ∈ X:

P0,i j (t, A) = I(i = j)q0,i (t, A) 2 ∫ t

+ P0,k j (t − s, A)Q0,ik (ds), t ≥ 0, i ∈ X, k=1

(3.109)

0

where, for t ≥ 0, i, j ∈ X, Q0,i j (t) = P{ζ0,1 ≤ t, η0,1 = j},

(3.110)

and, for A ∈ BZ, t ≥ 0, i, j ∈ X, q0,i (t, A) = Pi {ξ0 (t) ∈ A, ηε (t) = i, ζε,1 > t}.

(3.111)

In this case, it should be assumed that (a) the characteristics of the stochastic triplets ξ¯ε,uε ,i,n = ξε,uε ,n (t), t ≥ 0 , κε,uε ,i,n, ηε,uε ,i,n , i = 1, 2, n = 1, 2, . . . replace, for every ε ∈ (0, 1], the corresponding characteristics of the original stochastic triples ξ¯ε,i,n = ξε,i,n (t), t ≥ 0 , κε,i,n, ηε,i,n , i = 1, 2, n = 1, 2, . . ., in the conditions O2 –R2 , and (b) the characteristics of the limiting stochastic triplets ξ¯0,i,n = ξ0,i,n (t), t ≥ 0 , κ0,i,n, η0,i,n , i = 1, 2, n = 1, 2, . . . replace the corresponding limiting characteristics in the conditions O2 –R2 .

3.3.2 Aggregation of Regeneration Times and Embedded Regenerative Processes It is always possible to build for each ε ∈ (0, 1] new aggregated regeneration times, so that the alternating regenerative process (ξε (t), ηε (t)), t ≥ 0, becomes a standard regenerative process with these new regeneration times.

3.3 Time compression and aggregation of regeneration times

65

3.3.2.1 Stopping Times for the Embedded Markov Chain. Let us assume that the probabilities pε,12, pε,21 > 0. In what follows, we use three different types of stopping times for the Markov chains ηε,n . The first is the stopping time θ˜ε [r] = min(k > r : ηε,k  ηε,r ), which is, for r = 0, 1, . . ., the first after r time of change of the state ηε,r . The second is the stopping time θˆε [r] = min(k > r : ηε,k = ηε,r ), which is, for r = 0, 1, . . ., the first after r return time to the state ηε,r . The third is the stopping time θˇε [r] = min(k > θ˜ε [r] : ηε,k = ηε,r ), which is, for r = 0, 1, . . ., the first after θ˜ε [r] return time to the state ηε,r . Obviously, the return times θˆε [r] and θˇε [r] are connected by the inequalities r < θˆε [r] ≤ θˇε [r], for r = 0, 1, . . .. 3.3.2.2 Embedded Regenerative Processes. Let us define the stopping times ˆ νˆε,n−1 ], n = 1, 2, . . ., and νˇε,0 = 0, νˇε,n = θ[ ˇ νˇε,n−1 ], n = 1, 2, . . ., νˆε,0 = 0, νˆε,n = θ[ which are successive return times to the state ηε,0 for the Markov chain ηε,n . Let us also consider the successive return times ζˆε,n = ζε,νˆ ε, n , n = 0, 1, . . ., and ζˇε,n = ζε,νˇ ε, n , n = 0, 1, . . ., to the state ηε (0) for the semi-Markov process ηε (t). The process ξˆε (t) = (ξε (t), ηε (t)), t ≥ 0, is a regenerative process with regeneration times ζˆε,n, n = 0, 1, . . .. We call it embedded regenerative process of the first type.   = 0, νˆε,1 = We can also consider the shifted sequences of stopping times νˆε,0  = θ[  ˜θ ε [0], νˆε,n ˆ νˆ  ˆ  ε,n−1 ], n = 2, 3, . . ., and ζε,n = ζε,νˆ ε, n , n = 0, 1, . . .. ˆ The process ξε (t) = (ξε (t), ηε (t)), t ≥ 0, is a regenerative process with regenera , n = 0, 1, . . ., and a transition period [0, ζˆ  ). tion times ζˆε,n ε,1 The process ξˇε (t) = (ξε (t), ηε (t)), t ≥ 0, also is a regenerative process with regeneration times ζˇε,n , n = 0, 1, . . .. We call it the embedded regenerative process of the second type.   = 0, νˇε,1 = We can also consider the shifted sequences of stopping times νˇε,0  = θ[  ˇ νˇ  ˇ ˜θ ε [0], νˇε,n  ε,n−1 ], n = 2, 3, . . ., and ζε,n = ζε,νˇ ε, n , n = 0, 1, . . .. The process ξˇε (t) = (ξε (t), ηε (t)), t ≥ 0 also is a regenerative process with  , n = 0, 1, . . ., and a transition period [0, ζˇ ). regeneration times ζˇε,n ε,1 If ηε (0) = 1, then the stopping times ζˆε,n, n = 1, 2, . . ., and ζˇε,n, n = 1, 2, . . ., are successive return times to the state 1 for the semi-Markov process ηε (t). As far as the  and ζˇ  are concerned, ζˆ  = ζˇ  is the first hitting time shifted stopping times ζˆε,n ε,n ε,1 ε,1  to the state 2, while ζˇε,n, n = 2, 3, . . ., and ζˇε,n, n = 2, 3, . . ., are successive return times to the state 2 for the semi-Markov process ηε (t). If ηε (0) = 2, then the stopping times ζˆε,n, n = 1, 2, . . ., and ζˇε,n, n = 1, 2, . . ., are successive return times to the state 2 for the semi-Markov process ηε (t). As far as the  and ζˇ  are concerned, ζˆ  = ζˇ  is the first hitting time shifted stopping times ζˆε,n ε,n ε,1 ε,1 to the state 1, while ζˆε,n, n = 2, 3, . . ., and ζˇε,n, n = 2, 3, . . ., are successive return times to the state 1 for the semi-Markov process ηε (t).  work well in ergodic We shall see that aggregated regeneration times ζˆε,n and ζˆε,n theorems for models with regular perturbations. However, these regeneration times

66

3 Perturbed ARP

are not suitable for models with singular and super-singular perturbations. Here, the  should be used. regeneration times ζˇε,n and ζˇε,n 3.3.2.3 Embedded Standard Alternating Regenerative Processes. Let ν˜ε,0 = ˜ ν˜ε,n−1 ], n = 1, 2, . . ., be the corresponding successive moments of state 0, ν˜ε,n = θ[ change for the Markov chain ηε,n . Let also us consider successive moments ζ˜ε,n = ζε,ν˜ ε, n , n = 0, 1, . . ., of change of state by the semi-Markov process ηε (t). The process ξ˜ε (t) = (ξε (t), ηε (t)), t ≥ 0, also is a standard alternating regenerative process with the regeneration times ζ˜ε,n, n = 0, 1, . . .. Characterisation of the alternating regenerative process ξ˜ε (t) as “standard” one is related the simple structure of the modulating semi-Markov process ηε (t). If the initial values ηε (0) = i, then ηε (t) = i, for ζ˜ε,2n ≤ t < ζ˜ε,2n+1 and ηε (t) = j  i, for ζ˜ε,2n+1 ≤ t < ζ˜ε,2n+2 , for n = 0, 1, . . ..   = 0, ν˜ε,1 = We can also consider the shifted sequences of stopping times ν˜ε,0    ˜ νˆ ˜ε,n = ζε,ν˜  , n = 0, 1, . . .. ], n = 2, 3, . . ., and ζ θ˜ε [0], ν˜ε,n = θ[ ε, n ε,n−1 The process ξ˜ε (t) = (ξε (t), ηε (t)), t ≥ 0, is also an alternating regenerative process  , n = 0, 1, . . ., and a transition period [0, ζ˜  ). with regeneration times ζ˜ε,n ε,1 It is also worth noting that the stopping times νˇε,n = ν˜ε,2n, n = 0, 1, . . ., and, thus, the regeneration times ζˇε,n = ζ˜ε,2n, n = 0, 1, . . .. Also, the shifted stopping times  = ν˜  , νˇ  = ν˜  ˇ ˜ ˇ ˜ νˇε,1 ε,1 ε,n ε,2n+1, n = 2, 3, . . ., and ζε,1 = ζε,1, ζε,n = ζε,2n+1, n = 2, 3, . . .. In conclusion, let us also mention the case, where the probabilities pε,12 = 0, pε,21 ≥ 0. In this case, the only successive return times νˆε,r , ζˆε,r , r = 0, 1, . . .,  , ζˆ  , r = 0, 1, . . ., can be defined as above, if, respectively, η (0) = 1 or or νˆε,r ε ε,r ηε (0) = 2. Similarly, if pε,12 ≥ 0, pε,21 = 0, then the only successive return times  , ζˆ  , r = 0, 1, . . ., can be defined as above, if, respecνˆε,r , ζˆε,r , r = 0, 1, . . ., or νˆε,r ε,r tively, ηε (0) = 2 or ηε (0) = 1.

3.3.3 Embedded Regenerative Processes and Ergodic Theorems for Perturbed Alternating Regenerative Processes 3.3.3.1 Ergodic Theorems and Embedded Regenerative Processes of First Type. Embedded regenerative processes of the first type can be effectively used for obtaining ergodic theorems for regularly perturbed alternating regenerative processes. In this model, it is assumed that an alternating regenerative process (ξε (t), ηε (t)) with regeneration times ζε,n is defined, for ε ∈ (0, 1], and also that the conditions O2 –R2 hold for these processes. Then, for ε ∈ (0, 1], an embedded regenerative process ξˆε (t) = (ξε (t), ηε (t)) with regeneration times ζˆε,n is constructed, by using the algorithm described in Sect. 3.3.2.2.

3.3 Time compression and aggregation of regeneration times

67

The following relation connects the distributions of the above alternating regenerative and the embedded regenerative processes, for A ∈ BZ, i, j ∈ X, t ≥ 0, Pε,i j (t, A) = Pi {ξε (t) ∈ A, ηε (t) = j} = Pi { ξˆε (t) ∈ A × { j}}.

(3.112)

The idea is to prove that the conditions O1 –R1 hold for the embedded regenerative processes ξˆε (t) and, thus, the ergodic Theorems 2.1–2.3 can be applied to them. In this way, the corresponding ergodic relations for the probabilities Pε,i j (tε, A), for 0 ≤ tε → ∞ as ε → 0, can be obtained. Such theorems also can be extended to models, in which the conditions O2 – R2 do not hold for the alternating regenerative processes (ξε (t), ηε (t)) with the regeneration times ζε,n , but there exists a time compression factor uε > 0 such that the above conditions hold for the compressed in time alternating regenerative processes (ξε,uε (t), ηε,uε (t)) = (ξε (tuε ), ηε (tuε )) with the regeneration times ζε,uε ,n = u−1 ε ζε,n . In this case, the idea is to prove that the conditions O1 –R1 hold for the compressed in time embedded regenerative processes ξˆεuε (t) = ξˆε (tuε ) and, thus, ergodic Theorems 2.1–2.3 can be applied to them. In this way, the corresponding ergodic relations for the probabilities Pε,i j (tε uε, A), for 0 ≤ tε → ∞ as ε → 0, can be obtained. The construction based on the use of embedded regenerative processes of the first type works well for regularly perturbed alternating regenerative processes. As a matter of fact, the weak convergence of distribution functions of regeneration times and their first moments to the first moments of the corresponding limiting distribution functions for regeneration times is required in Theorems 2.1–2.3. As it was shown in Sects. 4.1.1.2 and 4.1.1.3, this simultaneous convergence holds for regularly perturbed embedded regenerative processes. However, the above-mentioned simultaneous convergence may not take place for singularly perturbed embedded regenerative processes. In this case, embedded regenerative processes of the second type may be used. 3.3.3.2 Ergodic Theorems and Embedded Regenerative Processes of Second Type. In this model, it is assumed that an alternating regenerative process (ξε (t), ηε (t)) with regeneration times ζε,n is defined, for ε ∈ (0, 1], and also that the conditions O2 –R2 hold for these processes. Then, for ε ∈ (0, 1] and some time compression factor vε > 0, a compressed in time embedded regenerative process ξˇε,vε (t) = (ξε (tvε ), ηε (tvε )) with regeneration times ζˇε,vε ,n = vε−1 ζε,n is constructed using the algorithm described in Sect. 3.3.2.2. The following relation connects the distributions of the above compressed in time alternating regenerative and the embedded regenerative processes, for A ∈ BZ, i, j ∈ X, t ≥ 0: Pε,i j (tvε, A) = Pi {ξε (tvε ) ∈ A, ηε (tvε ) = j} = Pi { ξˇε,vε (t) ∈ A × { j}}.

(3.113)

The idea is to prove that the conditions O1 –R1 hold for the embedded regenerative processes ξˇε,vε (t) and, thus, the ergodic Theorems 2.1–2.3 can be applied to them.

68

3 Perturbed ARP

In this way, the corresponding ergodic relations for the probabilities Pε,i j (tε vε, A) for 0 ≤ tε → ∞ as ε → 0 can be obtained. Such theorems also can be extended to models, in which the conditions O2 – R2 do not hold for the alternating regenerative processes (ξε (t), ηε (t)) with the regeneration times ζε,n , but there exists a time compression factor uε > 0 such that the above conditions hold for the compressed in time alternating regenerative processes (ξε,uε (t), ηε,uε (t)) = (ξε (tuε ), ηε (tuε )) with the regeneration times ζε,uε ,n = u−1 ε ζε,n . In this case, the idea is to prove that the conditions O1 –R1 hold for the compressed in time embedded regenerative processes ξˇε,vε uε (t) = ξˇε (tvε uε ) and, thus, the ergodic Theorems 2.1–2.3 can be applied to them. In this way, ergodic relations for the probabilities Pε,i j (tε vε uε, A), for 0 ≤ tε → ∞ as ε → 0, can be obtained. The construction based on the use of embedded regenerative processes of the second type works well for singularly perturbed alternating regenerative processes. As it is shown in Sect. 4.1.1.3, the weak convergence of distribution functions of regeneration times and their first moments to the first moments of the corresponding limiting distribution functions for regeneration times, which is required in Theorems 2.1–2.3, hold for embedded regenerative processes of the second type.

Chapter 4

Ergodic Theorems for Regularly Perturbed Alternating Regenerative Processes

In this chapter, we present individual ergodic theorems for perturbed alternating regenerative processes modulated by regularly perturbed two-state semi-Markov processes (shortly referred as regularly perturbed alternating regenerative processes). This is the case where both regime switching probabilities for perturbed modulating semi-Markov processes converge to positive limiting values, as ε → 0. We also present ergodic theorems for semi-regularly perturbed alternating regenerative processes. This is case where one regime switching probability for perturbed modulating semi-Markov processes converges to a positive limiting value, while another one converges to 0, as ε → 0. This chapter includes four sections. In Sect. 4.1, we consider regularly perturbed alternating regenerative processes and describe procedures of aggregation of regeneration times appropriate for the asymptotic ergodic analysis of such processes. In Sect. 4.2, we consider regularly perturbed standard alternating regenerative processes. The corresponding ergodic asymptotics is presented in Theorem 4.1. In Sect. 4.3, we consider regularly perturbed alternating regenerative processes. The corresponding ergodic asymptotics is presented in Theorem 4.2. In Sect. 4.4, we consider semi-regularly perturbed alternating regenerative processes. The corresponding ergodic asymptotics is presented in Theorems 4.3 and 4.4.

4.1 Regularly Perturbed Alternating Regenerative Processes and Embedded Regenerative Processes of the First Type In this section, we introduce models of regularly and semi-regularly perturbed alternating regenerative processes and the corresponding embedded regenerative processes of the first type.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0_4

69

70

4 Ergodic theorems for regularly perturbed ARP

4.1.1 Regularly and Semi-regularly Perturbed Alternating Regenerative Processes These are alternating regenerative processes with a regular perturbation model, where, in addition to O2 –R2 , the following condition holds: T: p0,12 + p0,21 > 0. The conditions T and O2 (a) imply that three cases are possible: (a) 0 < pε,12 → p0,12 as ε → 0, where p0,12 > 0 and 0 < pε,21 → p0,21 as ε → 0, where p0,21 > 0. (b) 0 < pε,12 → p0,12 as ε → 0, where p0,12 > 0 and 0 < pε,21 → p0,21 = 0 as ε → 0 or pε,21 ≡ 0. (c) 0 < pε,21 → p0,21 as ε → 0, where p0,21 > 0 and 0 < pε,12 → p0,12 = 0 as ε → 0 or pε,12 ≡ 0. The case of regularly perturbed processes is where p0,12, p0,21 > 0. The case of semi-regularly perturbed processes is where p0,12 > 0, p0,21 = 0 or p0,12 = 0, p0,21 > 0. In the case of regular or semi-regular perturbed processes, the probabilities pε,12 and pε,21 are asymptotically comparable in the sense that condition Sβ holds for some β ∈ [0, ∞], that is, the following relation holds: βε =

pε,12 p0,12 →β= ∈ [0, ∞] as ε → 0. pε,21 p0,21

(4.1)

In the case of regularly perturbed alternating regenerative processes, the parameter β ∈ (0, ∞), while in the case of semi-regularly perturbed alternating regenerative processes β = 0 or β = ∞. In both cases, the limiting embedded Markov chain η0,n with the matrix of transition probabilities p0,i j  is ergodic. In the case of the model with regular perturbations, the stationary probabilities 1 , α2 (β) = 1+β1 −1 . for the Markov chain η0,n are α1 (β) = 1+β In case of the model with semi-regular perturbations, the corresponding stationary probabilities for the Markov chain η0,n are α1 (0) = 1, α2 (0) = 0, if β = 0, or α1 (∞) = 0, α2 (∞) = 1, if β = ∞. In what follows in this section, we assume that (ξε (t), ηε (t)) is an alternating regenerative process with regeneration times ζε,n , for every ε ∈ (0, 1], and also that the conditions O2 –R2 and T are satisfied for these alternating regenerative processes.

4.1.2 Embedded Regenerative Processes of the First Type 4.1.2.1 Renewal Equations for Distributions of Embedded Regenerative Processes of the First Type. Let ξˆε, (t) = (ξε (t), ηε (t)) be the embedded regenerative process of the first type with regeneration times ζˆε,n constructed using the procedure described in Sect. 3.3.2.2.

4.1 Regularly perturbed ARP and embedded RP

71

Let us also assume that the initial state ηε (0) = i ∈ X, and the transition probabilities pε, ji > 0, j  i, for ε ∈ (0, 1]. In this case, Pi { ζˆε,n < ∞} = 1, n = 0, 1, . . .. The process ξˆε, (t) = (ξε (t), ηε (t)) has the phase space Z = Z×X. The σ-algebra of measurable sets BZ includes sets of the form B = (A1 ×{1})∪(A2 ×{2}), A1, A2 ∈ BZ . Let us consider the probabilities: Pˆε,i (t, B) = Pi { ξˆε (t) ∈ B} = Pi {(ξε (t), ηε (t) ∈ B} = Pi {(ξε (t), ηε (t)) ∈ A1 × {1}} + Pi {(ξε (t), ηε (t)) ∈ A2 × {2}} = Pε,i1 (t, A1 ) + Pε,i2 (t, A2 ), t ∈ R+, B ∈ BZ .

(4.2)

The idea behind considering embedded regenerative processes ζˆε (t) with regenerative times ζˆε,n is to show that the conditions O1 –R1 hold for these processes, then to apply the ergodic Theorems 2.1–2.3 to these processes, and, in this way, to obtain the corresponding ergodic relations for the probabilities Pˆε,i (tε, B) and Pε,i j (tε, A) as 0 ≤ tε → ∞ as ε → 0. The function Pˆε,i (·, B) is, for every B ∈ BZ , the unique in the class L solution for the following renewal equation: ∫ t (4.3) Pˆε,i (t − s, B)Fˆε,i (ds), t ≥ 0, Pˆε,i (t, B) = qˆε,i (t, B) + 0

where, for t ≥ 0,

Fˆε,i (t) = Pi { ζˆε,1 ≤ t},

(4.4)

and, for B ∈ BZ , t ≥ 0, qˆε,i (t, B) = Pi { ξˆε (t) ∈ B, ζˆε,1 > t}.

(4.5)

The function qˆε,i (t, B) belongs to the class P[BZ ]. Moreover, this function is consistent with the tail probability function 1 − Fˆε,i (·), that is, the following relation holds:  = 1 − Fˆε,i (t), for t ∈ R+ . qˆε,i (t, Z)

(4.6)

The tail probability function 1− Fˆε,i (t) and the function qˆε,i (t, B) can be expressed in terms of the transition probabilities Q ε,i j (t), the probability tail functions 1 − Fε, j (t), j = 1, 2, and the functions qε, j (t, A), j = 1, 2. Let Q(∗n) ε, j j (t) be n-fold convolution of the distribution function Q ε, j j (t) (it is a proper or improper distribution function) for n = 0, 1, . . . and j ∈ X. Let, also, Uε, j (t) be the corresponding renewal function defined by the following formula, for j ∈ X: ∞

Uε, j (t) = Q(∗n) (4.7) ε, j j (t), t ≥ 0. n=0

72

4 Ergodic theorems for regularly perturbed ARP

Under the above assumptions, Uε, j (t) < ∞, t ∈ R+ and it is a non-negative, non-decreasing, and continuous from the right function. Let us also define functions, for j  i: Uˆ ε,i j (t) = Q ε,i j (t) ∗ Uε, j (t) ∫ t Q ε,i j (ds)Uε, j (t − s), t ≥ 0. =

(4.8)

0

The following relation takes place, for t ∈ R+, j  i: 1 − Fˆε,i (t) = Pi { ζˆε,1 > t} ∞

= Pi {ζε,1 > t} + = 1 − Fε,i (t) +

Pi {ηε,r = j, 1 ≤ r ≤ n, ζε,n ≤ t < ζε,n+1 }

n=1 ∫ t 0

where Fε,i (t) =

Uˆ ε,i j (ds)(1 − Fε, j (t − s)),



Q ε,i j (t), t ≥ 0, i ∈ X.

(4.9)

(4.10)

j ∈X

Similarly, for t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}), A1, A2 ∈ BZ, j  i, qˆε,i (t, B) = Pi { ξˆε (t) ∈ B, ζˆε,1 > t} = Pi {ξε (t) ∈ Ai, ζε,1 > t} ∞

+ Pi {ξε (t) ∈ A j , ηε,r = j, 1 ≤ r ≤ n, ζε,n ≤ t < ζε,n+1 } n=1

= qε,i (t, Ai ) +

∫ 0

t

Uˆ ε,i j (ds)qε, j (t − s, A j ).

(4.11)

It is useful noting that Z = (Z × {1}) ∪ (Z × {2}) = Z × X. In this case, the relation (4.11) takes the following form consistent with the relation (4.6), for t ∈ R+, j  i: ∫ t  qˆε,i (t, Z) = qε,i (t, Z) + Uˆ ε,i j (ds)qε, j (t − s, Z) 0 ∫ t Uˆ ε,i j (ds)(1 − Fε, j (t − s)) = 1 − Fε,i (t) + 0

= 1 − Fˆε,i (t), for t ∈ R+ .

(4.12)

The relations (4.11) and (4.12) give a hint that the corresponding limiting distribution function Fˆ0,i (t) and the function qˆ0,i (t, B) appearing for the embedded regenerative processes ξˆε (t) in the conditions P1 and R1 may also be given by the following relations similar to (4.9) and (4.11), that is, for t ∈ R+, j  i:

4.1 Regularly perturbed ARP and embedded RP

∫

73 t

1 − Fˆ0,i (t) = 1 − F0,i (t) + 0

Uˆ 0,i j (ds)(1 − F0,u0, j (t − s)),

and, for t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}), A1, A2 ∈ BZ, j  i, ∫ t qˆ0,i (t, B) = q0,i (t, Ai ) + Uˆ 0,i j (ds)q0, j (t − s, A j ),

(4.13)

(4.14)

0

where, for j  i, U0, j (t) =

∞

Q(∗n) 0, j j (t), t ≥ 0,

(4.15)

n=0

and Uˆ 0,i j (t) = Q0,i j (t) ∗ U0, j (t) ∫ t Q0,i j (ds)U0, j (t − s), t ≥ 0. =

(4.16)

0

Since the limiting distribution function Fˆ0,i (·) should be proper, it also should be assumed that the limiting probability p0, ji > 0, for j  i. In this case, U0, j (t) < ∞, t ∈ R+ , and it is a non-negative, non-decreasing, and continuous from the right function. Note also that the function Fˆ0,i (·) given by the relation (4.13) is the distribution function of the first return time to the state i for a semi-Markov process ηˆ0 (t) with the phase space X and the transition probabilities Q0,i j (t). The function qˆ0,i (t, B), t ∈ R+, B ∈ BZ , given by relation (4.14) belongs to the class P[BZ ], for i ∈ X. It follows from the assumption that the condition R1 is satisfied for the alternating processes (ξε (t), ηε (t)) with regeneration times ζε,n . Thus, the functions q0, j (t, A), t ∈ R+, A ∈ BZ, j = 1, 2, also belong to the class P[BZ ]. Also, relations (4.13)–(4.15) obviously imply that the function qˆ0,i (t, B), t ∈ R+, B ∈ BZ , is consistent with the tail probability function 1 − Fˆ0,i (t), for i ∈ X. Let us also mention the case where the existence of a limiting alternating regenerative process (ξ0 (t), η0 (t)) with regeneration times ζ0,n is assumed and the corresponding embedded regenerative process ξˆ0 (t) with regeneration times ζˆ0,n is constructed using the procedure described in Sect. 3.3.2.2. In this case, it should be assumed that the transition probabilities Q0,i j (t), t ≥ 0, i, j ∈ X, appearing in the condition P2 , and the functions q0, j (t, A), t ≥ 0, A ∈ Γ, j = 1, 2, appearing in the condition R2 are defined via the corresponding limiting alternating regenerative process. In this case, the function qˆ0,i (t, B), t ∈ R+, B ∈ BZ , automatically belongs to the class P[BZ ] and it is consistent with the tail probability function 1 − Fˆ0,i (t). Of course, relations (4.13)–(4.15) also take place for the limiting distribution function Fˆ0,i (t) and function qˆ0,i (t, B). The above-described construction of embedded regenerative processes of the first type is used below for obtaining ergodic theorems for regularly and semi-

74

4 Ergodic theorems for regularly perturbed ARP

regularly perturbed alternating regenerative processes. These results are presented in Theorems 4.1–4.4 and 5.1, 5.2. 4.1.2.2 Compressed in Time Embedded Regenerative Processes of the First Type. In this case, it is assumed that, for every ε ∈ (0, 1], the conditions O2 – R2 are satisfied for the compressed in times alternating regenerative processes (ξε,uε (t), ηε,uε (t)) = (ξε (tuε ), ηε (tuε )) with the regeneration times ζε,uε ,n = u−1 ε ζε,n , where uε, ε ∈ (0, 1] is a positive function serving as a time compression factor. The quantities pε,i j , Q ε,i j (t), Fε,i (t), eε,i , and qε,i (t, A) defined for the alternating regenerative processes in Sect. 3.1.2 should be replaced, for every ε ∈ (0, 1], by the quantities pε,uε ,i j = pε,i j , Q ε,uε ,i j (t) = Q ε,i j (tuε ), Fε,uε ,i (t) = Fε,i (tuε ), eε,uε ,i = u−1 ε eε,i , and qε,uε ,i (t, A) = qε,i (tuε , A) in the conditions O2 –R2 , while the corresponding limiting quantities p0,u0,i j , Q0,i j (t), F0,i (t), e0,i , and q0,i (t, A) do not change in these conditions. In this case, the corresponding embedded regenerative process of the first type constructed with the use of procedure described in Sect. 3.3.2.2 is also compressed in ˆ time. It is the process ξˆε,uε (t) = ξˆε (tuε ) with the regeneration times ζˆε,uε ,n = u−1 ε ζε,n . The relation (4.2) takes in this case the following form: Pˆε,uε ,i (t, B) = Pi { ξˆε,uε (t) ∈ B} = Pi {(ξε (tuε ), ηε (tuε ) ∈ B} = Pi {(ξε (tuε ), ηε (tuε )) ∈ A1 × {1}} + Pi {(ξε (tuε ), ηε (tuε )) ∈ A2 × {2}} = Pε,i1 (tuε, A1 ) + Pε,i2 (tuε, A2 ), t ∈ R+, B ∈ BZ .

(4.17)

As above, the idea behind considering embedded regenerative processes ζˆε,uε (t) with regenerative times ζˆε,uε ,n is to show that the conditions O1 –R1 are satisfied for these processes, then to apply the ergodic Theorems 2.1–2.3 to these processes, and, in this way, to obtain the corresponding ergodic relations for the probabilities Pˆε,i (tε, B) and Pε,i j (tε uε, A) as 0 ≤ tε → ∞ as ε → 0. The above-described construction of compressed in time embedded regenerative processes of the first type for regularly and semi-regularly perturbed alternating regenerative processes is described below, in Sect. 5.2. This makes it possible to generalise the ergodic Theorems 4.1–4.4 and 5.1, 5.2 to the model of compressed in time regularly and semi-regularly perturbed alternating regenerative processes.

4.2 Ergodic Theorems for Perturbed Standard Alternating Regenerative Processes In this section, we present individual ergodic theorem for perturbed standard alternating regenerative processes. Let us consider regularly perturbed standard alternating regenerative processes, where, additionally to O2 –R2 , the following condition holds:

4.2 Perturbed standard ARP

75

T1 : pε,12, pε,21 = 1, for ε ∈ (0, 1]. The condition P2 (a) implies that, in this case, the limiting probabilities: p0,12, p0,21 = 1.

(4.18)

p0,12 = 1. p0,21

(4.19)

and, thus, the parameter: β=

The Markov chain η0,n is ergodic. Its stationary probabilities: α1 (1) = α2 (1) =

1 . 2

(4.20)

The conditions O2 –R2 and T1 imply that the semi-Markov process η0 (t) is also ergodic. Its stationary probabilities have the form: ρ1 (1) =

e0,1 e0,2 , ρ2 (1) = . e0,1 + e0,2 e0,1 + e0,2

(4.21)

The corresponding limiting stationary probabilities for the perturbed alternating regenerative process (ξε (t), ηε (t)) have the form: π0,(1)j (A) = ρ j (1)π0, j (A), for A ∈ BZ, j ∈ X.

(4.22)

The following ergodic theorem for regularly perturbed standard alternating regenerative processes takes place. Theorem 4.1 Let the conditions O2 –R2 and T1 be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0, Pε,i j (tε, A) → π0,(1)j (A) as ε → 0.

(4.23)

˜ = r + 1 and Proof In the case where the condition T1 holds, the stopping times θ[r] ˆθ[r] = r + 2, for r = 0, 1, . . ..   = 0, ζˆε,1 = Thus, the regeneration times ζˆε,n = ζε,2n, n = 0, 1, . . ., and ζˆε,0  ζε,1, ζˆε,n = ζε,2n−1, n = 2, 3, . . .. Therefore, the standard alternating regenerative process ξˆε (t) = (ξε (t), ηε (t)), t ≥ 0, is a standard regenerative process with regeneration times ζˆε,0 = ζε,0, ζˆε,1 = ζε,2, ζˆε,2 = ζε,4, . . .. It also can be considered as a regenerative process with transition  ) = [0, ζ ) and regenerative times ζˆ  ˆ ˆ period [0, ζε,1 ε,1 ε,0 = ζε,0, ζε,1 = ζε,1, ζε,2 =  ζε,3, ζˆε,3 = ζε,5, . . .. Regenerative lifetimes are not involved. We can use Theorems 2.1–2.3, for the model with the stopping probabilities qε = 0, ε ∈ [0, 1]. Let us analyse the asymptotic behaviour of the probabilities Pˆε,1 (t, B). In this case, we do prefer to consider ξˆε (t) = (ξε (t), ηε (t)), t ≥ 0, as a standard regenerative process with regeneration times 0 = ζˆε,0, ζˆε,1, . . ., and the initial state ηε (0) = 1.

76

4 Ergodic theorems for regularly perturbed ARP

The renewal equation (4.3) for the probabilities Pˆε,1 (t, B) takes the following form, for B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ : ∫ t Pˆε,1 (t, B) = qˆε,1 (t, B) + (4.24) Pˆε,1 (t − s, B)Fˆε,11 (ds), t ≥ 0, 0

where, for t ≥ 0, Fˆε,11 (t) = P1 { ζˆε,1 ≤ t} = P1 {ζε,2 ≤ t},

(4.25)

and, for B ∈ BZ , t ≥ 0, qˆε,1 (t, B) = P1 { ξˆε (t) ∈ B, ζˆε,1 > t} = P1 {(ξε (t), ηε (t)) ∈ B, ζε,2 > t}.

(4.26)

Obviously, ζˆε,n ≥ ζε,n , for n = 0, 1, . . .. Thus, the condition O2 implies that the condition O1 (a) is satisfied. The condition O1 (b) is also satisfied, since the stopping probability qε ≡ 0. In this case, for t ≥ 0, Fˆε,11 (t) = P1 {ζε,2 ≤ t} = Q ε,12 (t) ∗ Q ε,21 (t) = Fε,12 (t) ∗ Fε,21 (t),

(4.27)

and, thus, ∫ eˆε,11 =

0

∞

t Fˆε,11 (dt) = E1 ζε,2 = eε,12 + eε,21 .

(4.28)

Note that the condition T1 implies that the expectations eε,11, eε,22 = 0 and, therefore, (4.29) eˆε,11 = eε,12 + eε,21 = eε,1 + eε,2 . The relation (4.27) and the conditions P2 and T1 imply that the condition P1 (a) is satisfied for the distribution functions Fˆε,11 (·), that is, Fˆε,11 (·) = Fε,12 (·) ∗ Fε,21 (·) ⇒ Fˆ0,11 (·) = F0,12 (·) ∗ F0,21 (·) as ε → 0.

(4.30)

The relation (4.27) and the condition P2 (c) imply that the condition P1 (b) is satisfied. Indeed, let us introduce characteristic functions: ∫ ∞ ϕˆ0,11 (z) = eizu Fˆ0,11 (du), z ∈ R1, (4.31) 0

∫

and ϕ0,i j (z) =

0

∞

eizu F0,i j (du), z ∈ R1, i, j ∈ X.

(4.32)

4.2 Perturbed standard ARP

77

The relation (4.30) implies that ϕˆ0,11 (z) = ϕ0,12 (z)ϕε,21 (z) z  0.

(4.33)

The condition P2 (c) implies, by Lemma A.1, that |ϕ0,12 (z)|, |ϕε,21 (z)| < 1, for z  0. These relations and the relation (4.33) imply that | ϕˆ0,11 (z)| = |ϕ0,12 (z)||ϕε,21 (z)| < 1 for z  0.

(4.34)

Thus, by Lemma A.1, the distribution function Fˆ0,11 (·) generating the limiting renewal equation (4.24), for ε = 0, is non-arithmetic, i.e., condition P1 (b) is satisfied for this distribution function. The relation (4.28) and the condition Q2 imply that the condition Q1 is satisfied for the expectations eˆε,11 , that is, eˆε,11 = eε,12 + eε,21 → eˆ0,11 = e0,12 + e0,21 as ε → 0.

(4.35)

In this case, the function qˆε,1 (t, B) given by relation (4.11) takes the following form, for t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ : ∫ t Q ε,12 (ds)qε,2 (t, A2 ), (4.36) qˆε,1 (t, B) = qε,1 (t, A1 ) + 0

and the corresponding limiting function qˆ0,1 (t, B) given by relation (4.14) takes the following form, for t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ : ∫ t Q0,12 (ds)q0,2 (t, A2 ), (4.37) qˆ0,1 (t, B) = q0,1 (t, A1 ) + 0

where Q0,12 (t) and q0, j (t, A j ), i = 1, 2, are the corresponding limiting quantities appearing in the conditions P2 and R2 . As it was shown in Sect. 4.1.1.1, the condition R1 (a) is satisfied for the function qˆ0,1 (t, B), t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ . Consider the sets B = A × {1}, A ∈ Γ. In this case, for t ∈ R+ , Pˆε,1 (t, A × {1}) = P1 {ξε (t) ∈ A, ηε (t) = 1} = Pε,11 (t, A), and, thus, the renewal equation (4.24) takes the following form: ∫ t Pε,11 (t, A) = qˆε,11 (t, A) + Pε,11 (t − s, A)Fˆε,11 (ds), t ≥ 0, 0

where, for t ∈ R+, A ∈ BZ ,

(4.38)

(4.39)

78

4 Ergodic theorems for regularly perturbed ARP

qˆε,11 (t, A) = qˆε,1 (t, A × {1}) = P1 { ξˆε (t) ∈ A × {1}, ζˆε,1 > t} = P1 {ξε (t) ∈ A, ηε (t) = 1, ζε,2 > t}.

(4.40)

If ηε (0) = 1, then ηε (t) = 1, for t ∈ [0, ζε,1 ), and ηε (t) = 2, for t ∈ [ζε,1, ζε,2 ). Therefore, for every t ∈ R+, A ∈ BZ , qˆε,11 (t, A) = P1 {ξε (t) ∈ A, ηε (t) = 1, ζε,2 > t} = P1 {ξε (t) ∈ A, ζε,1 > t} = qε,1 (t, A).

(4.41)

The relation (4.41) and the condition R2 (b)–(d) imply that the conditions R1 (b)–(d) are satisfied for the functions qˆε,11 (·, A) = qε,1 (·, A), for A ∈ Γ, that is: us (1) the functions qˆε,11 (·, A) = qε,1 (·, A) −→ q0,1 (·, A) = qˆ0,11 (·, A) as ε → 0, for s ∈ U[q ·,1 (·, A)], A ∈ Γ, (2) m(U¯ [q ·,1 (·, A)]) = 0, A ∈ Γ, and (3) the function q0,1 (·, A) is continuous almost everywhere with respect to Lebesgue measure on B+ , for A ∈ Γ. As mentioned above, the stopping probabilities qε ≡ 0 and, thus, qε tε → 0 as ε → 0, for any 0 ≤ tε → ∞ as ε → 0. Thus, all conditions of Theorem 2.1 are satisfied and, thus, the ergodic relation given in this theorem holds for the probabilities Pε,11 (tε, A). In this case, it takes the form of the following asymptotic relation, which holds for any A ∈ Γ and 0 ≤ tε → ∞ as ε → 0: ∫ ∞ 1 qˆ0,11 (s, A)ds Pε,11 (tε, A) → eˆ0,11 0 ∫∞ q0,1 (s, A)ds e0,1 0 = e0,1 + e0,2 e0,1 (1) = ρ1 (1)π0,1 (A) = π0,1 (A) as ε → 0.

(4.42)

The relation (4.42) coincides with the relation (4.23), for the case i, j = 1. Second, let us analyse the asymptotic behaviour of the probabilities Pε,21 (t, A). In this case, we do prefer to consider (ξε (t), ηε (t)), t ≥ 0, as a regenerative process  ) and regenerative times ζ  , ζ  = ζ , ζ  , . . .. with transition period [0, ζε,1 ε,1 ε,2 ε,0 ε,1 The shifted process (ξε (ζε,1 + t), ηε (ζε,1 + t)), t ≥ 0, is a standard regenerative process. If ηε (0) = 2, then ηε (ζε,1 ) = 1. That is why, the probabilities Pε,11 (t, A) play for the above shifted regenerative process the role of the probabilities Pε(1) (t, A) defined in Sect. 2.1.1.2.  ) has, in The distribution function for the duration of the transition period [0, ζε,1 this case, the following form:  Fε,2 (·) = P2 {ζε,1 ≤ ·} = P2 {ζε,1 ≤ ·} = Q ε,21 (·).

The relation (4.43) and the conditions P2 and T1 imply that

(4.43)

4.3 Regularly perturbed ARP

79

Fε,2 (·) = Q ε,21 (·) ⇒ F0,2 (·) = Q0,21 (·) as ε → 0,

(4.44)

and, thus, the condition P 1 is satisfied. Therefore, all conditions of Theorem 2.2 hold, and, thus, the ergodic relation (4.42) for the probabilities Pε,11 (tε, A) also holds for the probabilities Pε,21 (tε, A), i.e., for A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0, (1) Pε,21 (tε, A) → π0,1 (A) as ε → 0.

(4.45)

Due to the symmetry of the conditions O2 –R2 , and T1 with respect to the indices i, j = 1, 2, the ergodic relations similar to above ergodic relations (4.42) and (4.45) for the probabilities Pε,11 (tε, A) and Pε,21 (tε, A) also take place for the probabilities (1) Pε,22 (tε, A) and Pε,12 (tε, A). In this case, the stationary probabilities π0,1 (A) should (1) (A), in these ergodic relations. be replaced by the stationary probabilities π0,2



Remark 4.1 The ergodic relation (4.23) given in Theorem 4.1 and the relation (4.17) imply that the following ergodic relation holds for any set B = (A1 × {1}) ∪ (A2 × {2}), A1, A2 ∈ Γ, i ∈ X and 0 ≤ tε → ∞ as ε → 0: Pˆε,i (tε, B) = Pi {(ξε (tε ), ηε (tε )) ∈ B} (1) (1) (A1 ) + π0,2 (A2 ) as ε → 0. → πˆ0(1) (B) = π0,1

(4.46)

4.3 Ergodic Theorems for Regularly Perturbed Alternating Regenerative Processes In this section, we present individual ergodic theorem for general regularly perturbed alternating regenerative processes. In this model, it is assumed that, additionally to O2 –R2 , the following condition is satisfied: T2 : p0,12, p0,21 > 0. Note that the condition T1 is a particular case of the condition T2 . In this case, the parameter β takes the following form: β=

p0,12 ∈ (0, ∞). p0,21

(4.47)

The Markov chain η0,n is ergodic, and its stationary probabilities: α1 (β) =

1 1 , α2 (β) = . 1+β 1 + β−1

(4.48)

The conditions O2 –Q2 and T2 imply that the semi-Markov process η0 (t) is ergodic. Its stationary probabilities have the form:

80

4 Ergodic theorems for regularly perturbed ARP

ρ1 (β) =

e0,1 α1 (β) e0,2 α2 (β) , ρ2 (β) = . e0,1 α1 (β) + e0,2 α2 (β) e0,1 α1 (β) + e0,2 α2 (β)

(4.49)

The corresponding limiting stationary probabilities for the regularly perturbed alternating regenerative processes (ξε (t), ηε (t)) have the following form, for β ∈ (0, ∞): (β) (4.50) π0, j (A) = ρ j (β)π0, j (A), for A ∈ BZ, j ∈ X. The following ergodic theorem for regularly perturbed alternating regenerative processes takes place. Theorem 4.2 Let the conditions O2 –R2 and T2 be satisfied, and the parameter p0,12 /p0,21 = β ∈ (0, ∞). Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0, (β) (4.51) Pε,i j (tε, A) → π0, j (A) as ε → 0. Proof As was pointed out in Sect. 3.3.2.2, the process ξˆε (t) = (ξε (t), ηε (t)) is a regenerative process with regeneration times ζˆε,n, n = 0, 1, . . .. It is also a regenerative  , n = 0, 1, . . ., and a transition period [0, ζˆ  ). process with regeneration times ζˆε,n ε,1 Again, regenerative lifetimes are not involved. We can use Theorems 2.1–2.3, for the model with the stopping probabilities qε = 0, ε ∈ [0, 1]. As above, let us first analyse the asymptotic behaviour of the probabilities Pˆε,1 (t, B). In this case, we do prefer to consider ξˆε (t) = (ξε (t), ηε (t)), t ≥ 0, as a standard regenerative process with regeneration times ζˆε,n, n = 0, 1, . . ., and the initial state ηε (0) = 1. The renewal equation (4.3) for the probabilities Pˆε,1 (t, B) takes the following form, for B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ : ∫ t ˆ (4.52) Pε,1 (t, B) = qˆε,1 (t, B) + Pˆε,1 (t − s, B)Fˆε,11 (ds), t ≥ 0, 0

where, for t ≥ 0, Fˆε,11 (t) = Fˆε,1 (t) = P1 { ζˆε,1 ≤ t}, t ≥ 0,

(4.53)

and, for A ∈ BZ, t ≥ 0, qˆε,1 (t, B) = P1 { ξˆε (t) ∈ B, ζˆε,1 > t} = P1 {(ξε (t), ηε (t)) ∈ B, ζˆε,1 > t}.

(4.54)

Obviously, ζˆε,n ≥ ζε,n , for n = 0, 1, . . .. Thus, the condition O2 implies that the condition O1 (a) is satisfied. The condition O1 (b) is also satisfied, since the stopping probabilities qε ≡ 0. In this case, Fˆε,11 (t) is the distribution function of the first return time to the state 1 for the semi-Markov process ηε (t). It can be expressed in terms of convolutions of the transition probabilities for this semi-Markov process. Namely, for t ≥ 0,

4.3 Regularly perturbed ARP

81

Fˆε,11 (t) = Q ε,11 (t) + Q ε,12 (t) ∗

∞

Q(∗n) ε,22 (t) ∗ Q ε,21 (t)

n=0

= Fε,11 (t)pε,11 + Fε,12 (t)pε,12 ∗

∞

∗n n Fε,22 (t)pε,22 ∗ Fε,21 (t)pε,21 .

(4.55)

n=0

The relation (4.55) takes the following equivalent form in terms of Laplace transforms: ∫ ∞ e−st Qˆ ε,11 (dt) φˆε,11 (s) = 0

= ψε,11 (s) + ψε,12 (s)

∞

n ψε,22 (s)ψε,21 (s)

n=0

ψε,12 (s)ψε,21 (s) 1 − ψε,22 (s) φε,12 (s)pε,12 φε,21 (s)pε,21 , s ≥ 0. = φε,11 (s)pε,11 + 1 − φε,22 (s)pε,22 = ψε,11 (s) +

(4.56)

The relation (4.55) implies that the random variable νˆε,1 has the so-called burned geometric distribution, that is,  1 with probability pε,11, νˆε,1 = n−2 p n with probability pε,12 pε,22 ε,21, for n ≥ 2. This fact and the conditions Q2 and T2 imply, in an obvious way, that the expectation eˆε,11 = E1 ζˆε,1 < ∞. It is easy to calculate, for example, using the first derivative of the Laplace transform φˆε,11 (s) at zero,  eˆε,11 = E1 ζˆε,1 = −φˆε,11 (0) 1 pε,21 = eε,11 + eε,12 1 − pε,22

eε,22 1 + pε,12 pε,21 + eε,21 2 1 − pε,22 (1 − pε,22 ) eε,1 pε,21 + eε,2 pε,12 = pε,21

=

eε,1 α1 (βε ) + eε,2 α2 (βε ) . α1 (βε )

(4.57)

The relation (4.56) and the conditions P◦2 (equivalent to the condition P2 ), T2 imply that, for s ≥ 0,

82

4 Ergodic theorems for regularly perturbed ARP

φˆε,11 (s) → φˆ0,11 (s) = φ0,11 (s)p0,11 +

ϕ0,12 (s)p0,12 ϕ0,21 (s)p0,21 as ε → 0. 1 − ϕ0,22 (s)p0,22

(4.58)

Therefore, the condition P1 (a) is satisfied for the distribution functions Fˆε,11 (·), that is, (4.59) Fˆε,11 (·) ⇒ Fˆ0,11 (·) as ε → 0, where Fˆ0,11 (·) = Q0,11 (·) + Q0,12 (·) ∗

∞

Q∗n 0,22 (·) ∗ Q 0,21 (·)

n=0

= F0,11 (t)p0,11 + F0,12 (t)p0,12 ∗

∞

∗n n F0,22 (t)p0,22 ∗ F0,21 (t)p0,21 .

(4.60)

n=0

Also, the relation (4.60) and the condition P2 (c) imply that the condition P1 (b) is satisfied. Indeed, let us introduce the characteristic function: ∫ ∞ ϕˆ0,11 (z) = eizu Fˆ0,11 (du), z ∈ R1 . (4.61) 0

According to the relation (4.60), this characteristic function is given by the following relation: ϕˆ0,11 (z) = ϕ0,11 (z)p0,11 +

ϕ0,12 (z)p0,12 ϕ0,21 (z)p0,21 , z ∈ R1 . 1 − ϕ0,22 (z)p0,22

(4.62)

Note that, according to the condition T2 , the probabilities p0,12, p0,21 > 0. The condition P2 (c) implies, by lemma A.1, that |ϕ0,12 (z)| < 1, |ϕ0,21 (z)| < 1, z  0. Thus, |ϕ0,12 (s)|p0,12 |ϕ0,21 (s)|p0,21 1 − |ϕ0,22 (s)|p0,22 |ϕ0,12 (s)|p0,12 |ϕ0,21 (s)|p0,21 ≤ p0,11 + 1 − p0,22 = p0,11 + |ϕ0,12 (s)||ϕ0,21 (s)|p0,12 < 1, for z  0.

| ϕˆ0,11 (z)| ≤ |ϕ0,11 (s)|p0,11 +

(4.63)

Therefore, by Lemma A.1, the distribution function Fˆ0,11 (·) is non-arithmetic. The relation (4.57) and the conditions P2 (which implies the fulfilment of the relation (3.64)), Q2 , and T2 imply that the condition Q1 is satisfied, that is,

4.3 Regularly perturbed ARP

83

eε,1 α1 (βε ) + eε,2 α2 (βε ) α1 (βε ) e0,1 α1 (β0 ) + e0,2 α2 (β0 ) as ε → 0. → eˆ0,11 = α1 (β0 )

eˆε,11 =

(4.64)

In this case, the function qˆε,1 (t, B) given by the relation (4.11) takes the following form, for t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ : ∫ t (4.65) qˆε,1 (t, B) = qε,1 (t, A1 ) + Uˆ ε,12 (ds)qε,2 (t, A2 ). 0

where the function Uˆ ε,12 (t) is given by the relation (4.8). The corresponding limiting function qˆ0,1 (t, B) given by the relation (4.14) takes the following form, for t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ : ∫ t (4.66) qˆ0,1 (t, B) = q0,1 (t, A1 ) + Uˆ 0,12 (ds)q0,2 (t, A2 ), 0

where the function Uˆ 0,12 (t) is given by the relation (4.16), and and Q0,1, j (t), q0, j (t, A j ), i = 1, 2, are the corresponding limiting quantities appearing in the conditions P2 and R2 . As it was shown in Sect. 4.1.1.1, the condition R1 (a) is satisfied for the function qˆ0,1 (t, B), t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ . Let us consider sets B = A × {1}, A ∈ Γ. In this case, for t ∈ R+ , Pˆε,1 (t, A × {1}) = P1 {ξε (t) ∈ A, ηε (t) = 1} = Pε,11 (t, A), and the renewal equation (4.24) takes the following form: ∫ t Pε,11 (t, A) = qˆε,11 (t, A) + Pε,11 (t − s, A)Fˆε,11 (ds), t ≥ 0,

(4.67)

(4.68)

0

where, for t ∈ R+, A ∈ BZ , qˆε,11 (t, A) = qˆε,1 (t, A × {1}) = P1 { ξˆε (t) ∈ A × {1}, ζˆε,1 > t} = P1 {ξε (t) ∈ A, ηε (t) = 1, ζˆε,1 > t}.

(4.69)

If ηε (0) = 1, then ηε (t) = 1 for t ∈ [0, ζε,1 ). Also, ζˆε,1 = ζε,1 , if ηε,1 = 1, and ηε (t) = 2, for t ∈ [ζε,1, ζˆε,1 ), if ηε,1 = 2. Therefore, for every A ∈ BZ, t ≥ 0, qˆε,11 (t, A) = P1 {ξε (t) ∈ A, ηε (t) = 1, ζˆε,1 > t} = P1 {ξε (t) ∈ A, ζε,1 > t} = qε,1 (t, A).

(4.70)

84

4 Ergodic theorems for regularly perturbed ARP

The relation (4.70) and the condition R2 (b)–(d) imply that the conditions R1 (b)–(d) are satisfied for the functions qˆε,11 (·, A), for A ∈ Γ, that is: (1) the functions us qˆε,11 (·, A) = qε,1 (·, A) −→ q0,1 (·, A) = qˆ0,11 (·, A) as ε → 0, for s ∈ U[q ·,1 (·, A)], A ∈ Γ, (2) m(U¯ [q ·,1 (·, A)]) = 0, A ∈ Γ, and (3) the function q0,1 (·, A) is continuous almost everywhere with respect to Lebesgue measure on B+ . As was mentioned above, the stopping probabilities qε ≡ 0 and, thus, qε tε → 0 as ε → 0, for any 0 ≤ tε → ∞ as ε → 0. Thus, all conditions of Theorem 2.1 hold, and the ergodic relation given in this theorem takes place for the probabilities Pε,11 (tε, A). In this case, it takes the form of the relation (4.51), where one should choose i, j = 1, i.e., for A ∈ Γ, and any 0 ≤ tε → ∞ as ε → 0, ∫ ∞ 1 qˆ0,11 (s, A)ds Pε,11 (tε, A) → eˆ0,11 0 ∫ ∞ α1 (β) q0,1 (s, A)ds = e0,1 α1 (β) + e0,2 α2 (β) 0 ∫∞ q0,1 (s, A)ds e0,1 α1 (β) 0 = e0,1 α1 (β) + e0,2 α2 (β) e0,1 (β)

= ρ j (β)π0, j (A) = π0,1 (A) as ε → 0.

(4.71)

Second, let us analyse the asymptotic behaviour of the probabilities Pε,21 (t, A). In this case, we do prefer to consider ξˆε (t) = (ξε (t), ηε (t)), t ≥ 0, as a regenerative  , ζˆ  , ζˆ  , . . . and a transition period [0, ζˆ  ). process with regenerative times ζˆε,0 ε,1 ε,2 ε,1  + t), η ( ζˆ  + t)), t ≥ 0, is a standard regenerative The shifted process (ξε (ζˆε,1 ε ε,1  ) = 1. That is why, the probabilities P process. If ηε (0) = 2, then ηε (ζˆε,1 ε,11 (t, A)

play for this process the role of the probabilities Pε(1) (t, A) defined in Sect. 2.1.1.2.  ) has, in The distribution function for the duration of the transition period [0, ζˆε,1 this case, the following form:  Fε,2 (t) = P2 { ζˆε,1 ≤ t}

=

∞

Q(∗n) ε,22 (t) ∗ Q ε,21 (t), t ≥ 0.

(4.72)

n=0

The relation (4.72) takes the following equivalent form in terms of Laplace transforms:

4.4 Semi-regularly perturbed ARP

85

φε,2 (s) = =

∫

∞

0 ∞

e−st Fε,2 (dt)

n n φε,22 (s)pε,22 φε,21 (s)pε,21

n=0

=

φε,21 (s)pε,21 , s ≥ 0. 1 − φε,22 (s)pε,22

(4.73)

The relations (4.72), (4.73) and the conditions P◦2 (equivalent to condition P2 ) and T2 imply that, for s ≥ 0, φε,2 (s) → φ0,2 (s) =

φ0,21 (s)p0,21 as ε → 0. 1 − φ0,22 (s)p0,22

(4.74)

Therefore, the condition P ◦1 (equivalent to the condition P 1 ) is satisfied. Thus, all conditions of Theorem 2.2 hold, and the corresponding ergodic relation for the probabilities Pε,11 (tε, A) also holds for the probabilities Pε,21 (tε, A), that is, for A ∈ Γ, and any 0 ≤ tε → ∞ as ε → 0, (β)

Pε,21 (tε, A) → π0,1 (A) as ε → 0.

(4.75)

Due to the symmetry of the conditions O2 –R2 and T2 with respect to the indices i, j = 1, 2, the ergodic relations, similar to the mentioned above ergodic relations for the probabilities Pε,11 (tε, A) and Pε,21 (tε, A), also take place for the probabilities (β) Pε,22 (tε, A) and Pε,12 (tε, A). In this case, the stationary probabilities π0,1 (A) should (β)

be replaced by the stationary probabilities π0,2 (A) in the corresponding ergodic relations.  Remark 4.2 Theorem 4.1 is a particular case of Theorem 4.2. In this case, the ergodic relation (4.51) takes the form of ergodic relation (4.23).

4.4 Ergodic Theorems for Semi-regularly Perturbed Alternating Regenerative Processes In this section, we present individual ergodic theorem for general semi-regularly perturbed alternating regenerative processes. In this model, it is assumed that, additionally to O2 –R2 , the following condition is satisfied: T3 : (a) p0,12 = 0, p0,21 > 0, or (b) p0,12 > 0, p0,21 = 0.

86

4 Ergodic theorems for regularly perturbed ARP

If the condition T3 (a) is satisfied, the parameter β = 0. If the condition T3 (b) is satisfied, the parameter β = ∞. In this case, the Markov chain η0,n is also ergodic. If the condition T3 (a) is satisfied, the stationary probabilities of this Markov chain, (4.76) α1 (0) = 1, α2 (0) = 0. If the condition T3 (b) is satisfied, the stationary probabilities of this Markov chain, (4.77) α1 (∞) = 0, α2 (∞) = 1. The conditions O2 –Q2 and T3 imply that the semi-Markov process η0 (t) is ergodic. If the condition T3 (a) is satisfied, the stationary probabilities of this semi-Markov process: e0,i αi (0) = I(i = 1), i = 1, 2. (4.78) ρi (0) = e0,1 α1 (0) + e0,2 α2 (0) If the condition T3 (b) is satisfied, the stationary probabilities of this semi-Markov process: e0,i αi (∞) = I(i = 2), i = 1, 2. (4.79) ρi (∞) = e0,1 α1 (∞) + e0,2 α2 (∞) The corresponding limiting stationary probabilities for the semi-regularly perturbed alternating regenerative processes (ξε (t), ηε (t)) have the form, for β = 0, ∞: (β)

π0, j (A) = ρ j (β)π0, j (A), for A ∈ BZ, j ∈ X, that is, π0,(0)j (A)

 =

and π0,(∞) j (A) =



(4.80)

π0,1 (A) for A ∈ BZ, j = 1, 0 for A ∈ BZ, j = 2,

(4.81)

0 for A ∈ BZ, j = 1, π0,2 (A) for A ∈ BZ, j = 2.

(4.82)

The following ergodic theorems for semi-regularly perturbed alternating regenerative processes takes place. Theorem 4.3 Let the conditions O2 –R2 and T3 (a) be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0, Pε,i j (tε, A) → π0,(0)j (A) as ε → 0.

(4.83)

Theorem 4.4 Let the conditions O2 –R2 and T3 (b) be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0, Pε,i j (tε, A) → π0,(∞) j (A) as ε → 0.

(4.84)

4.4 Semi-regularly perturbed ARP

87

Proof The process ξˆε (t) = (ξε (t), ηε (t)) is a standard regenerative process with regeneration times ζˆε,n, n = 0, 1, . . .. It also is a regenerative process with regenerative  , n = 0, 1, . . ., and a transition period [0, ζˆ  ). times ζˆε,n ε,1 Again, regenerative lifetimes are not involved. We can use Theorems 2.1 and 2.2, for the model with the stopping probabilities qε = 0, ε ∈ [0, 1]. Let us consider the case where the condition T3 (a) is satisfied. Let us analyse the asymptotic behaviour of the probabilities Pε,1 (t, B). In this case, we prefer to consider (ξε (t), ηε (t)), t ≥ 0, as a standard regenerative process with regeneration times ζˆε,0, ζˆε,1, ζˆε,2, . . ., and the initial state ηε (0) = 1. Let us analyse the asymptotic behaviour of the probabilities Pε,11 (t, A). The renewal equation (4.3) for the probabilities Pˆε,1 (t, B) takes the following form, for B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ : ∫ t ˆ (4.85) Pε,1 (t, B) = qˆε,1 (t, B) + Pˆε,1 (t − s, B)Fˆε,11 (ds), t ≥ 0, 0

where, for t ≥ 0, j = 1, 2, Fˆε,11 (t) = Fˆε,1 (t) = P1 { ζˆε,1 ≤ t},

(4.86)

and, for A ∈ BZ, t ≥ 0, j ∈ X, qˆε,1 (t, B) = P1 { ξˆε (t) ∈ B, ζˆε,1 > t} = P1 {(ξε (t), ηε (t)) ∈ B, ζˆε,1 > t}.

(4.87)

Obviously, ζˆε,n ≥ ζε,n , for n = 0, 1, . . .. Thus, the condition O2 implies that the condition O1 (a) is satisfied. The condition O1 (b) is also satisfied, since the stopping probabilities qε ≡ 0. As pointed out in the comments related to the relation (4.55), Fˆε,11 (t) is the distribution function of the first return time to the state 1 for the semi-Markov process ηε (t), and the following formula similar to (4.56) takes place for its Laplace transform: ∫ ∞ e−st Fˆε,11 (dt) φˆε,11 (s) = 0

ψε,12 (s)ψε,21 (s) 1 − ψε,22 (s) φε,12 (s)pε,12 φε,21 (s)pε,21 , s ≥ 0. = φε,11 (s)pε,11 + 1 − φε,22 (s)pε,22

= ψε,11 (s) +

(4.88)

Also, the following formula similar to (4.57) takes place for expectations: eε,1 α1 (βε ) + eε,2 α2 (βε )  . (0) = eˆε,11 = E1 ζˆε,1 = −φˆε,11 α1 (βε )

(4.89)

88

4 Ergodic theorems for regularly perturbed ARP

The conditions P◦2 (equivalent to the condition P2 ) and T3 (a) imply that, in the relation (4.88), either ψε,12 (s) = 0, for s ≥ 0, if pε,12 = 0, for ε ∈ [0, 1], or ψε,12 (s) → 0 as ε → 0, for s ≥ 0, if 0 < pε,12 → 0 as ε → 0. Therefore, for s ≥ 0, φˆε,11 (s) → φˆ0,11 (s) = ψ0,11 (s) = φ0,11 (s) as ε → 0.

(4.90)

Therefore, the condition P1 (a) is satisfied, with the corresponding limiting distribution function: Fˆ0,11 (·) = F0,11 (·), that is, Fˆε,11 (·) = Fε,11 (·) ⇒ Fˆ0,11 (·) = F0,11 (·) as ε → 0.

(4.91)

The condition P2 (c) directly implies that the condition P1 (b) is satisfied for the distribution function Fˆ0,11 (·) = F0,11 (·). Similarly, in the relation (4.89), either eε,12 = 0, if pε,12 = 0, for ε ∈ [0, 1], or eε,12 → 0 as ε → 0, if 0 < pε,12 → 0 as ε → 0. It follows from this remark and the conditions O2 –Q2 that eε,1 α1 (βε ) + eε,2 α2 (βε ) α1 (βε ) → eˆ0,11 = e0,11 as ε → 0.

eˆε,11 =

(4.92)

Note also that the condition T3 (a) implies that the expectation e0,11 = e0,1 . Thus, the condition Q1 is satisfied, with the corresponding limiting expectation eˆ0,11 = e0,1 . In this case the function qˆε,1 (t, B) given by the relation (4.11) takes the following form, for t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ , ∫ t qˆε,1 (t, B) = qε,1 (t, A1 ) + (4.93) Uˆ ε,12 (ds)qε,2 (t, A2 ), 0

where the function Uˆ ε,12 (·) is given by the relation (4.8). The corresponding limiting function qˆ0,1 (t, B) given by the relation (4.14) takes the following form: for t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ , ∫ t qˆ0,1 (t, B) = q0,1 (t, A1 ) + Uˆ 0,12 (ds)q0,2 (t, A2 ) 0

= q0,1 (t, A1 ),

(4.94)

since, in this case, Q0,12 (·) ≡ 0 and, thus, the function Uˆ 0,12 (·) given by the relation (4.16) coincides with the function 0(·) ≡ 0. As it was shown in Sect. 13.1.1.1, the condition R1 (a) is satisfied for the function qˆ0,1 (t, B), t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ . Let us consider sets B = A × {1}, A ∈ Γ. In this case, for t ∈ R+ , Pˆε,1 (t, A × { j}) = P1 {ξε (t) ∈ A, ηε (t) = j} = Pε,1j (t, A),

(4.95)

4.4 Semi-regularly perturbed ARP

89

and the renewal equation (4.24) takes the following form: ∫ t Pε,1j (t, A) = qˆε,1j (t, A) + Pε,1j (t − s, A)Fˆε,11 (ds), t ≥ 0,

(4.96)

0

where, for t ∈ R+, A ∈ BZ, j = 1, 2, qˆε,1j (t, A) = qˆε,1 (t, A × { j}) = P1 { ξˆε (t) ∈ A × { j}, ζˆε,1 > t} = P1 {ξε (t) ∈ A, ηε (t) = j, ζˆε,1 > t}.

(4.97)

First, let us analyse the asymptotic behaviour of the probabilities Pε,11 (t, A). If ηε (0) = 1, then ηε (t) = 1 for t ∈ [0, ζε,1 ). Also, ζˆε,1 = ζε,1 , if ηε,1 = 1, and ηε (t) = 2, for t ∈ [ζε,1, ζˆε,1 ), if ηε,1 = 2. Therefore, the relation similar to (4.70) takes place, for every A ∈ BZ, t ≥ 0, qˆε,11 (t, A) = P1 {ξε (t) ∈ A, ηε (t) = 1, ζˆε,1 > t} = P1 {ξε (t) ∈ A, ζε,1 > t} = qε,1 (t, A).

(4.98)

The relation (4.98) and the condition R2 (b)–(d) imply that the conditions R1 (b)– (d) hold for the functions qˆε,11 (·, A) = qε,1 (·, A), for A ∈ Γ, that is: (1) the functions us qˆε,11 (·, A) = qε,1 (·, A) −→ q0,1 (·, A) = qˆ0,11 (·, A) as ε → 0, for s ∈ U[q ·,1 (·, A)], A ∈ Γ, (2) m(U¯ [q ·,1 (·, A)]) = 0, A ∈ Γ, and (3) the function q0,1 (·, A) is continuous almost everywhere with respect to Lebesgue measure on B+ . As was mentioned above, the stopping probabilities qε ≡ 0, and, thus, qε tε → 0 as ε → 0, for any 0 ≤ tε → ∞ as ε → 0. Thus, all conditions of Theorem 2.1 hold, and the ergodic relation given in this theorem takes place for the probabilities Pε,11 (tε, A). In this case, it takes the form of the relation (4.83), where one should choose i, j = 1, i.e., for any A ∈ Γ and 0 ≤ tε → ∞ as ε → 0: ∫ ∞ 1 qˆ0,11 (s, A)ds Pε,11 (tε, A) → eˆ0,11 0 ∫ ∞ 1 q0,1 (s, A)ds = e0,11 0 (0) = π0,1 (A) = π0,1 (A) as ε → 0.

(4.99)

Now, let us analyse the asymptotic behaviour of the probabilities Pε,12 (t, A). If ηε (0) = 1, then ηε (t) = 1 for t ∈ [0, ζε,1 ), and ζˆε,1 = ζε,1 , if ηε,1 = 1. Also, ηε (t) = 2, for t ∈ [ζε,1, ζˆε,1 ), if ηε,1 = 2. Therefore, for A ∈ BZ, t ≥ 0,

90

4 Ergodic theorems for regularly perturbed ARP

qˆε,12 (t, A) = P1 {ξε (t) ∈ A, ηε (t) = 2, ζˆε,1 > t} = P1 {ξε (t) ∈ A, ηε,1 = 2, ζε,1 ≤ t < ζˆε,1 } ≤ P1 {ζε,1 ≤ t, ηε,1 = 2} ≤ pε,12,

(4.100)

and, thus, for A ∈ BZ , sup qˆε,12 (t, A) ≤ pε,12 → 0 as ε → 0. t ≥0

(4.101)

The relation (4.101) and the condition T3 (a) imply that the conditions R1 (b)–(d) are satisfied for the functions qˆε,1 (·, A × {2}) = qˆε,12 (·, A), for A ∈ Γ, that is: (4) us the functions qˆε,12 (·, A) −→ qˆ0,12 (·, A) = 0(·) ≡ 0 as ε → 0, for s ∈ U[qˆ ·,12 (·, A)] = R+, A ∈ Γ, (5) m(U¯ [qˆ ·,12 (·, A)] = 0, A ∈ Γ, (6) the function 0(·) is continuous almost everywhere with respect to Lebesgue measure on B+ . Thus, all conditions of Theorem 2.1 are satisfied, and the ergodic relation given in this theorem takes place for the probabilities Pε,12 (tε, A). In this case, it takes the form of the relation (4.83), where one should choose i = 1, j = 2, i.e., for any A ∈ Γ and 0 ≤ tε → ∞ as ε → 0, ∫ ∞ 1 qˆ0,12 (s, A)ds Pε,12 (tε, A) → eˆ0,11 0 ∫ ∞ 1 (0) = 0(s)ds = 0 = π0,2 (A) as ε → 0. (4.102) e0,11 0 Third, let us analyse the asymptotic behaviour of the probabilities Pε,2j (t, A), j = 1, 2. In this case, we prefer to consider (ξε (t), ηε (t)), t ≥ 0, as a regenerative process  , ζˆ  , ζˆ  , ζˆ  , . . . and a transition period [0, ζˆ  ). with regenerative times ζˆε,0 ε,1 ε,2 ε,3 ε,1  + t), η ( ζˆ  + t)), t ≥ 0, is a standard regenerative The shifted process (ξε (ζˆε,1 ε ε,1  ) = 1. That is why, the probabilities P process. If ηε (0) = 2, then ηε (ζˆε,1 ε,1j (t, A) play

for this process the role of the probabilities Pε(1) (t, A) defined out in Sect. 12.1.2.1.  ≤ t} and the Laplace transform The distribution function Fε,2 (t) = P2 { ζˆε,1 ∫∞ −st  ) are  ˜ φε,12 (s) = 0 e Q ε,12 (dt) for the duration of the transition period [0, ζˆε,1 given in relations (4.72) and (4.73), respectively. These relations and the conditions P◦2 and T3 (a) imply that, for s ≥ 0, φε,2 (s) → φ0,2 (s) as ε → 0,

(4.103)

where the limiting Laplace transform φ0,2 (s) is given by relation (4.74). Therefore, the condition P ◦1 (equivalent to the condition P 1 ) is satisfied. Thus, all conditions of Theorem 2.2 hold, and the ergodic relations (4.99) and (4.102) for the probabilities Pε,1j (tε, A), j = 1, 2 also hold for the probabilities Pε,2j (tε, A), j = 1, 2. Due to symmetry of the conditions O2 –R2 and T3 with respect to the indices i, j = 1, 2, the corresponding asymptotic analysis for the probabilities

4.4 Semi-regularly perturbed ARP

91

Pε,i j (tε, A), i, j = 1, 2 (under the assumption of holding the condition T3 (b)) is similar to the asymptotic analysis for the probabilities Pε,i j (tε, A), i. j = 1, 2 (under the assumption that the condition T3 (a) is satisfied). The corresponding ergodic relation (4.84) takes place for the above probabilities, under the assumption that the  condition T3 (b) is satisfied.

Chapter 5

Ergodic Theorems for Regularly Perturbed Alternating Regenerative Processes Compressed in Time

In this chapter, we present individual ergodic theorems for regularly perturbed alternating regenerative processes compressed in time. This chapter includes two sections. In Sect. 5.1, we consider regularly perturbed alternating regenerative processes with degenerating regeneration times. The corresponding ergodic asymptotics is presented in Theorems 5.1 and 5.2. In Sect. 5.2, we consider regularly perturbed alternating regenerative processes compressed in time. We present more general perturbation conditions in Lemma 5.1, which let us generalise the ergodic Theorems 4.1–4.4 and 5.1, 5.2 for regularly perturbed alternating regenerative processes on the model of regularly perturbed alternating regenerative processes compressed in time.

5.1 Regularly Perturbed Alternating Regenerative Processes with Degenerating Regeneration Times In this section, we present ergodic theorems for regularly perturbed alternating regenerative processes with degenerating regeneration times. In this model, either the distribution function F0,1j (·), j in YY0,1 , or F0,2j (·), j ∈ Y0,2 are degenerate, i.e., coincide with the distribution function F0 (u) = I(u ≥ 0), which has a unit jump at the point 0. In two theorems formulated below, the condition T is assumed to be satisfied. This condition covers all cases of regularly and semi-regularly perturbed alternating regenerative processes, where one of the conditions T1 , T2 , or T3 is satisfied. It is convenient to reformulate the condition T in the following equivalent form: T4 : (a) p0,21 > 0, or (b) p0,12 > 0. The condition T4 (a) implies that the parameter β = β = 0, if p0,12 = 0, while β ∈ (0, ∞), if p0,12 > 0.

p0,12 p0,21

∈ [0, ∞). Moreover,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0_5

93

94

5 Ergodic theorems for regularly perturbed ARP compressed in time

Similarly, the condition T4 (b) implies that the parameter β ∈ (0, ∞]. Moreover, β = ∞, if p0,21 = 0, while β ∈ (0, ∞), if p0,21 > 0. In this case, the Markov chain η0,n is ergodic, and the stationary probabilities of this Markov chain are given by the following formulas: α1 (β) =

1 1 , α2 (β) = . 1+β 1 + β−1

(5.1)

The corresponding variants of the condition P2 take two different forms: P2 : (a) pε,i j → p0,i j as ε → 0, for i, j ∈ X, (b) Q ε,i j (·) ⇒ Q0,i j (·) as ε → 0, for i, j ∈ X, where Q0,i j (·) is a proper or improper distribution function such that Q0,i j (∞) = p0,i j , for i, j ∈ X, (c) F0,1j (·) = p−1 0,1j Q 0,1j (·) is a weakly non-arithmetic distribution function, for j ∈ Y0,1 , while F0,2j (·) = p−1 0,2j Q 0,2j (·) = I(· ≥ 0), for j ∈ Y0,2 or P2: (a) pε,i j → p0,i j as ε → 0, for i, j ∈ X, (b) Q ε,i j (·) ⇒ Q0,i j (·) as ε → 0, for i, j ∈ X, where Q0,i j (·) is a proper or improper distribution function such that Q0,i j (∞) = p0,i j , for i, j ∈ X, (c) F0,2j (·) = p−1 0,2j Q 0,2j (·) is a weakly non-arithmetic distribution function, for j ∈ Y0,2 , while F0,1j (·) = p−1 0,1j Q 0,1j (·) = I(· ≥ 0), for j ∈ Y0,1 . Let us re-call the Laplace transforms, for i, j ∈ X and ε ∈ (0, 1], ∫ ∞ ψε,i j (s) = e−su Q ε,i j (du), s ≥ 0.

(5.2)

0

Conditions P2 and P2 can be formulated in the equivalent form using the above Laplace transforms: P2◦ : (a) pε,i j → p0,i j as ε → 0, for i, j = 1, 2, (b) ψε,i j (s) → ψ0,i j (s) as ε → 0, ∫∞ for s ≥ 0 and i, j ∈ X, where ψ0,i j (s) = 0 e−su Q0,i j (du), s ≥ 0 is the Laplace transform of some proper or improper distribution such that Q0,i j (∞) = p0,i j , ∫∞ −su F for i, j ∈ X, (c) φ0,1j (s) = p−1 ψ (s) = e 0,1j (du), s ≥ 0 is the Laplace 0,1j 0,1j 0 transform of a weakly non-arithmetic distribution function F0,1j (·) = p−1 0,1j Q 0,1j (·), ψ (s) = 1, s ≥ 0, for j ∈ Y for j ∈ Y0,1 , while φ0,2j (s) = p−1 0,2 0,2j 0,1j and P2◦ : (a) pε,i j → p0,i j as ε → 0, for i, j = 1, 2, (b) ψε,i j (s) → ψ0,i j (s) as ε → 0, ∫∞ for s ≥ 0 and i, j ∈ X, where ψ0,i j (s) = 0 e−su Q0,i j (du), s ≥ 0 is the Laplace transform of some proper or improper distribution such that Q0,i j (∞) = p0,i j , ∫∞ −su F for i, j ∈ X, (c) φ0,2j (s) = p−1 ψ (s) = e 0,2j (du), s ≥ 0 is the Laplace 0,2j 0,2j 0 transform of a weakly non-arithmetic distribution function F0,2j (·) = p−1 0,2j Q 0,2j (·), −1 for j ∈ Y0,2 , while φ0,1j (s) = p0,1j ψ0,1j (s) = 1, s ≥ 0, for j ∈ Y0,1 .

5.1 Regularly perturbed ARP with degenerating regeneration times

95

We assume, as before, that the condition O2 is satisfied and, thus, the alternating regenerative process (ξε (t), ηε (t)) is well defined on the time interval [0, ∞), for ε ∈ (0, 1]. The condition P2 is used in the case, where the condition T4 (a) is satisfied, while the condition P2 is used in the case, where the condition T4 (b) is satisfied. In both P

cases, the random variables ζ0,n −→ ∞ as n → ∞, and, thus, the limiting alternating regenerative process (ξ0 (t), η0 (t)) is, also, well defined on the time interval [0, ∞). It is useful to note that, under the condition P2 ,

Q0,2j (u) F0,2 (u) = j ∈Y0,2

=



F0,2j (u)p0,2j

j ∈Y0,2

= I(u ≥ 0)



p0,2j = I(u ≥ 0), for u ≥ 0.

(5.3)

j ∈Y0,2

Analogously, under the condition P2, F0,1 (u) = I(u ≥ 0), for u ≥ 0.

(5.4)

It is also useful noting that the condition Q2 and the relations (5.3) and (5.4) imply that, under the condition P2 ,

e0,2j p0,2j = 0, (5.5) e0,2 = j ∈Y0,2

and, under the condition P2, e0,1 =



e0,1j p0,1j = 0.

(5.6)

j ∈Y0,1

It is also useful noting that the assumption that the distribution functions F0,1j (·), j ∈ Y0,1 , or F0,2j (·), j ∈ Y0,2 are non-arithmetic used in the condition P2 is replaced in the conditions P2 and P2 by the weaker assumption that these distribution functions are weakly non-arithmetic. The following two theorems take place. Theorem 5.1 Let the conditions O2 , P2 , Q2 , R2 , and T4 (a) be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0, Pε,i j (tε, A) → π0,(0)j (A) as ε → 0.

(5.7)

Theorem 5.2 Let the conditions O2 , P2, Q2 , R2 , and T4 (b) be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0,

96

5 Ergodic theorems for regularly perturbed ARP compressed in time

Pε,i j (tε, A) → π0,(∞) j (A) as ε → 0.

(5.8)

Proof Let us consider the case, where the condition T4 (a) is satisfied and prove Theorem 5.1. The process ξˆε (t) = (ξε (t), ηε (t)) is a standard regenerative process with regeneration times ζˆε,n, n = 0, 1, . . .. It also is a regenerative process with regenerative times  , n = 0, 1, . . . and a transition period [0, ζˆ  ). ζˆε,n ε,1 Again, regenerative lifetimes are not involved. We can use Theorems 2.1 and 2.2, for the model with the stopping probabilities qε = 0, ε ∈ [0, 1]. The renewal equation (4.3) for the probabilities Pˆε,1 (t, B) takes the following form, for B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ : ∫ t ˆ (5.9) Pε,1 (t, B) = qˆε,1 (t, B) + Pˆε,1 (t − s, B)Fˆε,1 (ds), t ≥ 0, 0

where, for t ≥ 0, j = 1, 2, Fˆε,1 (t) = P1 { ζˆε,1 ≤ t},

(5.10)

and, for A ∈ BZ, t ≥ 0, j ∈ X, qˆε,1 (t, B) = P1 { ξˆε (t) ∈ B, ζˆε,1 > t} = P1 {(ξε (t), ηε (t)) ∈ B, ζˆε,1 > t}.

(5.11)

Obviously, ζˆε,n ≥ ζε,n , for n = 0, 1, . . .. Thus, the condition O2 implies that the condition O1 (a) is satisfied. The condition O1 (b) is also satisfied, since, the stopping probabilities qε ≡ 0. The distribution function Fˆε,11 (u) takes, in this case, the following form, for t ≥ 0: Fˆε,1 (t) = Fε,11 (t)pε,11 + Fε,12 (t)pε,12 ∗

∞

∗n n Fε,22 (t)pε,12 ∗ Fε,21 (t)pε,21 .

(5.12)

n=0

The Laplace transform of the distribution function Fˆε,1 (u) takes the following form: φˆε,1 (s) = φε,11 (s)pε,11 + φε,12 (s)pε,12

φε,21 (s)pε,21 , for s ≥ 0. 1 − φε,22 (s)pε,22

(5.13)

The conditions T (a) and P2 imply that φε,21 (s)pε,21 → p0,21 as ε → 0, for s ≥ 0, since p0,21 > 0 and, thus, the state 1 ∈ Y0,2 . Also, φε,22 (s)pε,22 → p0,22 as ε → 0, for s ≥ 0, if the state 2 ∈ Y0,2 , i.e., p0,22 > 0, or φε,22 (s)pε,22 → 0 as ε → 0, for ¯ 0,2 , i.e., p0,22 = 0. s ≥ 0, if the state 2 ∈ Y

5.1 Regularly perturbed ARP with degenerating regeneration times

97

Also, the conditions T (a) and P2 imply that φε,1j (s)pε,1j → φ0,1j (s)p0,1j as ε → 0, for s ≥ 0, if the state j ∈ Y0,1 , while φε,1j (s)pε,1j → 0 as ε → 0, for s ≥ 0, ¯ 0,1 . if the state j ∈ Y The above asymptotic relations imply that, under the conditions T (a) and P2 , φε,21 (s)pε,21 → 1 as ε → 0, for s ≥ 0 1 − φε,22 (s)pε,22

(5.14)

and φˆε,1 (s) → φˆ0,1 (s) = φ0,1 (s) =



φ0,1j (s)p0,1j as ε → 0, for s ≥ 0.

(5.15)

j ∈Y0,1

Therefore, the condition P1 (a) is satisfied, with the corresponding limiting distribution function, Fˆ0,1 (·) = F0,1 (·), that is, Fˆε,1 (·) ⇒ Fˆ0,1 (·) = F0,1 (·) =



F0,1j (·)p0,1j as ε → 0.

(5.16)

j ∈Y0,1

Theorem 2.1 requires that the distribution function F0,1 (u) is weakly nonarithmetic. According to the condition P2 (b), the distribution functions F0,1j (·), j ∈ Y0,1 are assumed to be weakly non-arithmetic, i.e., for j ∈ Y0,1 , ∞

(F0,1j (nh) − F0,1j (nh − 0)) < 1, for h > 0.

(5.17)

n=0

The relation (5.17) implies that, ∞

n=0

(F0,1 (nh) − F0,1 (nh − 0)) ∞



(F0,1j (nh) − F0,1j (nh − 0)) p0,1j < 1, for h > 0. = j ∈Y0,1

(5.18)

n=0

Therefore, the distribution function Fˆ0,1 (·) is weakly non-arithmetic. Let us denote, for i ∈ X and ε ∈ [0, 1], ∫ ∞ eˆε,i = u Fˆ0,i (du). 0

(5.19)

98

5 Ergodic theorems for regularly perturbed ARP compressed in time

Note also, that the condition Q2 and the relations (5.5) and (5.13) imply that, the following relation holds: eˆε,1 = eε,11 + eε,12 +

eε,21 + eε,22 pε,21

→ eˆ0,1 = e0,11 + e0,12 + = e0,1 +

e0,21 + e0,22 p0,21

e0,2 = e0,1 as ε → 0. p0,21

(5.20)

In this case, the function qˆε,1 (t, B) given by the relation (4.11) takes the following form, for t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ : ∫ t (5.21) qˆε,1 (t, B) = qε,1 (t, A1 ) + Uˆ ε,12 (ds)qε,2 (t, A2 ), 0

where the function Uˆ ε,12 (·) is given by the relation (4.8). The corresponding limiting function qˆ0,1 (t, B) given by the relation (4.14) takes the following form, for t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ : ∫ t qˆ0,1 (t, B) = q0,1 (t, A1 ) + Uˆ 0,12 (ds)q0,2 (t, A2 ) 0

= q0,1 (t, A1 ),

(5.22)

since, in this case, the function, q0,2 (t, A) ≤ 1 − F0,2 (t) = 0, t ≥ 0, and, thus, the function Uˆ 0,12 (·) given by the relation (4.16) coincides with the function 0(·) ≡ 0. As it was shown in Sect. 4.1.2, the condition R1 (a) is satisfied for the function qˆ0,1 (t, B), t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ . Let us consider sets B = A × {1}, A ∈ Γ. In this case, for t ∈ R+ , Pˆε,1 (t, A × {1}) = P1 {ξε (t) ∈ A, ηε (t) = j} = Pε,1j (t, A), and the renewal equation (4.24) takes the following form: ∫ t Pε,11 (t, A) = qˆε,1j (t, A) + Pε,11 (t − s, A)Fˆε,11 (ds), t ≥ 0, 0

where, for t ∈ R+, A ∈ BZ, j = 1, 2,

(5.23)

(5.24)

5.1 Regularly perturbed ARP with degenerating regeneration times

qˆε,11 (t, A) = qˆε,1 (t, A × {1}) = P1 { ξˆε (t) ∈ A × {1}, ζˆε,1 > t} = P1 {ξε (t) ∈ A, ηε (t) = 1, ζˆε,1 > t}.

99

(5.25)

If ηε (0) = 1, then ηε (t) = 1 for t ∈ [0, ζε,1 ). Also, ζˆε,1 = ζε,1 , if ηε,1 = 1, and ηε (t) = 2, for t ∈ [ζε,1, ζˆε,1 ), if ηε,1 = 2. Therefore, for every A ∈ BZ, t ≥ 0, qˆε,1 (t, A) = P1 {ξε (t) ∈ A, ηε (t) = 1, ζˆε,1 > t} = P1 {ξε (t) ∈ A, ζε,1 > t} = qε,1 (t, A).

(5.26)

The relation (5.26) and the condition R2 (b)–(d) imply that the conditions R1 (b)–(d) are satisfied for the functions qˆε,11 (·, A) = qε,1 (·, A), for A ∈ Γ, that is: us (1) the functions qˆε,11 (·, A) = qε,1 (·, A) −→ q0,1 (·, A) = qˆ0,11 (·, A) as ε → 0, for s ∈ U[q ·,1 (·, A)], A ∈ Γ, (2) m(U¯ [q ·,1 (·, A)]) = 0, A ∈ Γ, (3) the function q0,1 (·, A) is continuous almost everywhere with respect to Lebesgue measure on B+ . As was mentioned above, the stopping probabilities qε ≡ 0, and, thus, qε tε → 0 as ε → 0, for any 0 ≤ tε → ∞ as ε → 0. Thus, the conditions O1 –R1 of Theorem 2.1 are satisfied for the regenerative processes (ξε (t), ηε (t)), t ≥ 0 with regeneration times ζˆε,n, n = 0, 1, . . .. By applying this theorem, we get the following ergodic relation, for A ∈ Γ, and any 0 ≤ tε → ∞ as ε → 0: (5.27) Pε,11 (tε, A) → π0,1 (A) as ε → 0.  + t), η ( ζˆ  + t)), t ≥ 0 is a standard regenerative The shifted process (ξε (ζˆε,1 ε ε,1  ) = 1, where ζˆ  = ζ˜ . The time interval [0, ζˆ  ) process. If ηε (0) = 2, then ηε (ζˆε,1 ε,1 ε,1 ε,1 is a transition period for this regenerative process. The probabilities Pε,11 (t, A) play for this process the role of the probabilities Pε(1) (t, A) defined in Sect. 2.1.1.2.  ) has, in The distribution function for the duration of the transition period [0, ζˆε,1 this case, the following form:  Fε,2 (t) = P2 { ζˆε,1 ≤ t} ∞

= Q(∗n) ε,22 (t) ∗ Q ε,21 (t) n=0

=

∞

(∗n) n Fε,22 (t)pε,22 ∗ Fε,21 (t)pε,21, t ≥ 0.

(5.28)

n=0

Accordingly, the Laplace transform of the distribution function Fε,2 (t) takes the following form: φε,21 (s)pε,21 ψ ε,2 (s) = , for s ≥ 0. (5.29) 1 − φε,22 (s)pε,22 According to the relation (5.14),

100

5 Ergodic theorems for regularly perturbed ARP compressed in time

and, thus,

ψ ε,21 (s) → 1 as ε → 0 for s ≥ 0

(5.30)

Fε,2 (·) ⇒ F0,2 (·) = I(· ≥ 0).

(5.31)

The relation (5.31) implies that, in the case where ηε (0) = 2, the condition P 1 is satisfied. Thus, all conditions of Theorem 2.2 hold, and the ergodic relation (5.27) for the probabilities Pε,11 (tε, A) also holds for the probabilities Pε,21 (tε, A), i.e., for A ∈ Γ, and any 0 ≤ tε → ∞ as ε → 0, Pε,21 (tε, A) → π0,1 (A) as ε → 0.

(5.32)

The relations (5.27) and (5.32) imply also that, for A ∈ Γ, i = 1, 2, and any 0 ≤ tε → ∞ as ε → 0, Pε,i2 (tε, A) ≤ Pε,i2 (tε, Z) = 1 − Pε,i1 (tε, Z) → 1 − π0,1 (Z) = 0 as ε → 0. The proof of Theorem 5.1 is complete. The proof of Theorem 5.2 is similar.

(5.33) 

Remark 5.1 It is useful noting that in Theorem 5.1, i.e., in the case, where the conditions T4 (a) and P2 are satisfied, the condition R2 can be weakened. In fact, the assumption of the locally uniform convergence for the functions qε,i (t, A) given in this condition can be required to hold only for i = 1. As a matter of fact, the functions, qε,2 (t, A) ≤ P{κε,2,1 > t}, t ≥ 0 and, thus, by the condition P2 , the ut

functions qε,2 (·, A) −→ 0(·) ≡ 0 as ε → 0, for any t ∈ [0, ∞). Analogously, in Theorem 5.2, i.e., in the case where the conditions T4 (b) and P2 are satisfied, the condition R2 can be weakened and the assumption of the locally uniform convergence for the functions qε,i (t, A) given in this condition can be required only for i = 2. Remark 5.2 The condition T4 (a) is equivalent to the condition T4 plus the assumption that the parameter β ∈ [0, ∞). Thus, the case, where the condition T4 is satisfied and the parameter β = ∞, i.e., p0,12 > 0, p0,21 = 0, is excluded in Theorem 5.1. It is so, because in this case, the condition P2 is not enough for getting the asymptotic relations (5.16) and (5.20). Moreover, the condition P2 implies, in this case, that the distribution functions of regeneration times Fˆε,2 (·) ⇒ I(· ≥ 0) as ε → 0. Therefore, Theorem 2.2 cannot be applied. Remark 5.3 Analogously, the condition T4 (b) is equivalent to the condition T4 plus the assumption that the parameter β ∈ (0, ∞]. Thus, the case, where the condition T4 is satisfied and the parameter β = 0, i.e., p0,12 = 0, p0,21 > 0, is excluded in Theorem 5.2. It is so, because in this case, the condition P2 is not enough for getting the asymptotic relations similar to (5.16) and (5.20). Moreover, the condition P2

5.2 Compression in time for regularly perturbed ARP

101

implies that the distribution functions of regeneration times Fˆε,1 (·) ⇒ I(· ≥ 0) as ε → 0. Therefore, Theorem 2.2 cannot be applied.

5.2 Compression in Time for Regularly Perturbed Alternating Regenerative Processes In this section, we present a model of regularly perturbed alternating regenerative processes compressed in time and formulate corresponding more general perturbation conditions. We also describe a procedure of time compression that allows us to extend this model and to obtain more general ergodic theorems. Let us assume that the condition O2 is satisfied, and, also, that the following perturbation conditions, more general than conditions P2 and Q2 , are satisfied for the perturbed alternating regenerative processes (ξε (t), ηε (t)) with regenerative times ζε,n : Pˆ 2 : (a) pε,i j → p0,i j as ε → 0, for i, j ∈ X, (b) Q ε,i j (· uε,i ) ⇒ Q0,i j (·) as ε → 0, for i, j ∈ X, where Q0,i j (·) is a proper or improper distribution function such that Q0,i j (∞) = p0,i j , for i, j ∈ X, (c) F0,i j (·) = p−1 0,i j Q 0,i j (·) is a non-arithmetic distribution function, for j ∈ Y0,i, i ∈ X, (d) uε,i ∈ (0, ∞), ε ∈ (0, 1], i ∈ X and uε,i → u0,i ∈ (0, ∞] as ε → 0, for i ∈ X and ∫ ˆ 2 : (a) eε,i j < ∞, for ε ∈ (0, 1] and i, j ∈ X; (b) eε,i j /uε,i → e0,i j = ∞ tQ0,i j (dt) Q 0 < ∞ as ε → 0, for i, j ∈ X. The conditions Pˆ 2 (a) and (b) imply that the following relation holds, for i ∈ X:

Fε,i (· uε,i ) =

j ∈Y0, i

⇒



Q ε,i j (· uε,i ) +

Q ε,i j (· uε,i )

¯ 0, i j ∈Y



Q0,i j (·) = F0,i (·) as ε → 0.

(5.34)

j ∈Y0, i

Obviously, for i ∈ X, F0,i (·) =



Q0,i j (·) =

j ∈Y0, i



F0,i j (·)p0,i j .

(5.35)

j ∈Y0, i

Let, for ε ∈ (0, 1] and A ∈ BZ, i ∈ X, qε,i (tuε,i, A) = P{ξε,i (tuε,i ) ∈ A, ζε,i,1 > tuε,i ), t ≥ 0, i ∈ X,

(5.36)

where the regeneration process ξε,i (t) and its regeneration times ζε,i,n are defined by the relations (3.49) and (3.50). The function qε,i (·uε,i, A), t ∈ R+, A ∈ BZ belongs to the class P[BZ ] and it is consistent with the distribution function Fε,i (·uε,i ), for i ∈ X and ε ∈ (0, 1], i.e.,

102

5 Ergodic theorems for regularly perturbed ARP compressed in time

qε,i (tuε,i, Z) = 1 − Fε,i (tuε,i ), for t ∈ R+ .

(5.37)

We also assume that the following perturbation condition, more general than the condition R2 , is satisfied: ˆ 2 : There exist functions q0,i (t, A), t ≥ 0, A ∈ BZ , for i ∈ X, which belong to class R P[BZ ], a class of set Γ ⊆ BZ , and Borel sets U[q ·,i (· u ·,i, A)], A ∈ Γ, i ∈ X such that: (a) function q ·,i (t, A), t ∈ R+, A ∈ BZ is consistent with the distribution us function F0,i (·), for i ∈ X; (b) functions qε,i (· uε,i, A) −→ q0,i (·, A) as ε → 0, for points s ∈ U[q ·,i (· u ·,i, A)], A ∈ Γ, i ∈ X; (c) m(U¯ [q ·,i (· u ·,i, A)]) = 0, for A ∈ Γ, i ∈ X; (d) function q0,i (·, A) is continuous almost everywhere with respect to the Lebesgue measure m(·) on B+ , for A ∈ Γ, i ∈ X. It is also helpful to make a few comments about the above perturbation condition. It is worth noting that the notation U[q ·,i (· u ·,i, A)] is used to show that this set of convergence is, actually, determined by the family of functions qε,i (· uε,i, A), ε ∈ (0, 1]. The consistency relation (5.37) implies that, for every ε ∈ (0, 1] and A ∈ BZ, i ∈ X, the function qε,i (·uε,i, A) is majorised by the tail probability function 1−Fε,i (·uε,i ) on interval [0, ∞), that is, qε,i (tuε,i, A) ≤ 1 − Fε,i (tuε,i ), for t ∈ R+ .

(5.38)

In light of the relations (5.37) and (5.38), the consistency assumption regarding ˆ 2 , is natural. In this case, the family of measures q0,i (·, A), t ∈ R+ , made in condition R the following majorisation relation similar to (5.38) takes place, for A ∈ BZ, i ∈ X: q0,i (t, A) ≤ 1 − F0,i (t), for t ∈ R+ .

(5.39)

We also assume that the time compression factors uε,i, i ∈ X are asymptotically comparable in the sense that the following condition is satisfied: Xγ : uε,2 /uε,1 → γ ∈ [0, ∞] as ε → 0. Let us now introduce a time compression factor, for ε ∈ (0, 1], uε = uε,1 + uε,2 .

(5.40)

The condition Pˆ 2 (d) implies that, uε → u0 = u0,1 + u0,2 ∈ (0, ∞] as ε → 0.

(5.41)

Let ξ¯ε,uε ,i,n = ξε,uε ,n (t), t ≥ 0 , κε,uε ,i,n, ηε,uε ,i,n , i = 1, 2, n = 1, 2, . . ., be, for ε ∈ (0, 1], stochastic triplets constructed according to the time compression relation (3.96). Let also (ξε,uε (t), ηε,uε (t)) = (ξε (tuε ), ηε (tuε )), t ≥ 0 be, for ε ∈ (0, 1], a compressed in time alternating regenerative process with regeneration times ζε,uε ,n = u−1 ε ζε,n, n = 0, 1, . . . defined by the relations (3.97)–(3.99).

5.2 Compression in time for regularly perturbed ARP

103

The transition characteristics of the compressed in time alternating regenerative process (ξε,uε (t), ηε,uε (t)) are connected with the transition characteristics of the original alternating regenerative process (ξε (t), ηε (t)) by the following relations, for ε ∈ (0, 1]: Q ε,uε ,i j (t) = Q ε,i j (tuε ), t ≥ 0, i, j ∈ X, pε,uε ,i j = pε,i j = Q ε,i j (∞), i, j ∈ X, ∫ ∞ −1 −1 eε,uε ,i j = uε eε,i j = uε uQ ε,i j (du), i, j ∈ X, 0

qε,uε ,i (t, A) = qε,i (tuε, A), t ≥ 0, A ∈ BZ, i ∈ X.

(5.42)

The conditions Pˆ 2 and Xγ imply that the following relations take place, for i, j ∈ X: pε,uε ,i j = pε,i j → p0,i j as ε → 0,

(5.43)

and Q ε,uε ,i j (·) = Q ε,i j (· uε,i

uε ) uε,i

(γ)

⇒ Q0,i j (·) as ε → 0,

(5.44)

where ⎧ Q0,1j (· (1 + γ)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ p0,1j I(· ≥ 0) ⎪ (γ) Q0,i j (·) = p0,2j I(· ≥ 0) ⎪ ⎪ Q0,2j (· (1 + γ −1 )) ⎪ ⎪ ⎪ ⎪ 0 ⎩

for γ ∈ [0, ∞), j ∈ Y0,1, i = 1, for γ = ∞, j ∈ Y0,1, i = 1, for γ = 0, j ∈ Y0,2, i = 2, for γ ∈ (0, ∞], j ∈ Y0,2, i = 2, ¯ 0,i, i = 1, 2. for j ∈ Y

(5.45)

Note that, for j ∈ Y0,i, i = 1, 2 and γ ∈ [0, ∞], (γ)

Q0,i j (∞) = p0,i j .

(5.46)

Therefore, from the conditions Pˆ 2 , Xγ and the relation (5.44) it follows that the condition P2 , P2 , or P2 is satisfied (for the alternating regenerative processes (ξε,uε (t), ηε,uε (t)), t ≥ 0 with the regeneration times ζε,uε ,n, n = 0, 1, . . .), with the (γ) corresponding limiting distributions Q0,i j (·), i, j ∈ X, if, respectively, the parameter γ = 0, γ ∈ (0, ∞), or γ = ∞, Moreover, in two extremal cases ,where γ = 0 or γ = ∞, the assumption that, (0) (∞) (·), j ∈ Y0,1 or F0,2j (·), j ∈ Y0,2 are nonrespectively, the distribution functions F0,1j arithmetic can be replaced by the weaker assumption that these distribution functions are weakly non-arithmetic. ˆ 2 and Xγ imply that the following relation takes place, for The conditions Q i, j ∈ X:

104

5 Ergodic theorems for regularly perturbed ARP compressed in time

eε,uε ,i j = where (γ)

e0,i j

eε,i j eε,i j uε,i (γ) = → e0,i j as ε → 0, uε uε,i uε

⎧ e0,1j (1 + γ)−1 ⎪ ⎪ ⎪ ⎪ ⎪0 ⎨ ⎪ = 0 ⎪ ⎪ ⎪ e0,2j (1 + γ −1 )−1 ⎪ ⎪ ⎪ 0 ⎩

for γ ∈ [0, ∞), j ∈ Y0,1, i = 1, for γ = ∞, j ∈ Y0,1, i = 1, for γ = 0, j ∈ Y0,2, i = 2, for γ ∈ (0, ∞], j ∈ Y0,2, i = 2, ¯ 0,i, i = 1, 2. for j ∈ Y

(5.47)

(5.48)

ˆ 2 , Xγ and the relation (5.47) imply that, for the Therefore, the conditions Q alternating regenerative processes (ξε,uε (t), ηε,uε (t)), t ≥ 0 with the regeneration times ζε,uε ,n, n = 0, 1, . . ., the condition Q2 is satisfied, with the corresponding (γ) limiting expectations e0,i j , i, j ∈ X. Let us define the sets, U[γ, q ·,u·,i (· , A)]

⎧ {s ≥ 0 : s(1 + γ) ∈ U[q ·,i (· u · ,i, A)]} ⎪ ⎪ ⎪ ⎨ (0, ∞) ⎪ = (0, ∞) ⎪ ⎪ ⎪ ⎪ {s ≥ 0 : s(1 + γ −1 ) ∈ U[q ·,i (· u · ,i, A)]} ⎩

for γ for γ for γ for γ

∈ [0, ∞), i = 1, = ∞, i = 1, = 0, i = 2, ∈ (0, ∞], i = 2.

(5.49)

Obviously, m(U¯ [γ, q ·,u·,i (· , A)]) = 0, for γ ∈ [0, ∞], i ∈ X, A ∈ Γ, since, by the ˆ 2 , m(U¯ [q ·,i (· u · ,i, A)]) = 0, for i ∈ X, A ∈ Γ. condition R ˆ 2 and Xγ imply that the following relation takes place, for The conditions R s ∈ U[γ, q ·,u·,1 (· , A)], γ ∈ [0, ∞), A ∈ Γ: qε,uε ,1 (·, A) = qε,1 (· uε, A) us uε = qε,1 (· uε,1 , A) −→ q0,1 (·(1 + γ), A) as ε → 0, uε,1

(5.50)

and, for s ∈ U[γ, q ·,u·,2 (· , A)], γ ∈ (0, ∞], A ∈ Γ, qε,uε ,2 (·, A) = qε,2 (· uε, A) us uε , A) −→ q0,2 (· (1 + γ −1 ), A) as ε → 0. = qε,2 (· uε,i uε,i

(5.51)

ˆ 2 , Xγ and the relations (5.34), (5.38), (5.44) imply that, for Also, the conditions R s ∈ (0, ∞), γ = ∞ if i = 1 or γ = 0 if i = 2, and A ∈ Γ, qε,uε ,i (s, A) ≤ 1 − Fε,i (suε ) = 1 − Fε,i (suε,i

uε ) → 0 as ε → 0. uε,i

(5.52)

5.2 Compression in time for regularly perturbed ARP

105

ˆ 2 and Xγ imply that the following relation takes place, for Thus, the conditions R s ∈ U[γ, q ·,u·,i (· , A)], i = 1, 2, γ ∈ [0, ∞], A ∈ Γ: qε,uε ,i (·, A) = qε,i (· uε, A) us uε (γ) , A) −→ q0,i (·, A) as ε → 0, = qε,i (· uε,i uε,i

(5.53)

where (γ) q0,i (·,

⎧ q0,1 (·(1 + γ), A) ⎪ ⎪ ⎪ ⎨ 0(·) ≡ 0 ⎪ A) = 0(·) ≡ 0 ⎪ ⎪ ⎪ ⎪ q0,2 (·(1 + γ −1 ), A) ⎩

for γ for γ for γ for γ

∈ [0, ∞), i = 1, A ∈ BZ, = ∞, i = 1, A ∈ BZ, = 0, i = 2, A ∈ BZ, ∈ (0, ∞], i = 2, A ∈ BZ .

(5.54)

ˆ 2 , Xγ and the relation (5.50) imply that, for the alterTherefore, the conditions R nating regenerative processes (ξε,uε (t), ηε,uε (t)), t ≥ 0 with the regeneration times ζε,uε ,n, n = 0, 1, . . ., the condition R2 is satisfied, with the corresponding limiting (γ) functions q0,i (·, A), for γ ∈ [0, ∞], i ∈ X, A ∈ BZ . It is also useful noting that, according to the relations (5.48) and (5.54), the corresponding limiting stationary probabilities, ∫ ∞ 1 (γ) (γ) q0,i (s, A)m(ds) π0,i (A) = (γ) e0,i 0 ∫ ∞ 1 q0,i (s, A)m(ds) = π0,i (A), (5.55) = e0,i 0 do not depend on the parameter γ, in the cases, γ ∈ [0, ∞), i = 1, A ∈ BZ and γ ∈ (0, ∞], i = 2, A ∈ BZ . The following lemma summarises the above remarks. ˆ 2, R ˆ 2 , T4 , and Xγ (for some γ ∈ [0, ∞]) be Lemma 5.1 Let the conditions O2 , Pˆ 2 , Q satisfied for the alternating regenerative processes (ξε (t), ηε (t)). Then, the following propositions take place for the compressed in time alternating regenerative processes (ξε,uε (t), ηε,uε (t)) with the time compression factor uε given by the relation (5.40): (i) If γ ∈ (0, ∞), the conditions O2 , P2 , Q2 , and R2 are satisfied. (ii) If γ = 0, the conditions O2 , P2 , Q2 , and R2 are satisfied. (iii) If γ = ∞, the conditions O2 , P2, Q2 , and R2 are satisfied. (iv) The asymptotic relations (5.43), (5.44), (5.47), and (5.53) play the roles of the asymptotic relations appearing in the above conditions. The corresponding limiting quantities and sets are given by relations (5.42)–(5.55). Remark 5.4 In two cases, where the parameter γ = 0 or γ = ∞, the condition Pˆ 2 (c) can be weakened and replaced by the assumption that the distribution function

106

5 Ergodic theorems for regularly perturbed ARP compressed in time

F0,1j (·) is a weakly non-arithmetic, for j ∈ Y0,1 , if γ = 0, or by the assumption that the distribution function F0,2j (·) is a weakly non-arithmetic, for j ∈ Y0,2 , if γ = ∞. Remark 5.5 Lemma 5.1 allows to apply to the alternating regenerative processes ˆ 2, R ˆ 2, (ξε,uε (t), ηε,uε (t)) Theorems 4.2–4.4 and 5.1, 5.2, if the conditions O2 , Pˆ 2 , Q T4 , and Xγ are satisfied. In particular, Theorem 4.2 can be applied if γ ∈ (0, ∞), β ∈ (0, ∞), Theorem 4.3 if γ ∈ (0, ∞), β = 0, Theorem 4.4 if γ ∈ (0, ∞), β = ∞, Theorem 5.1 if γ = 0, β ∈ [0, ∞), and Theorem 5.2 if γ = ∞, β ∈ (0, ∞]. Here, β is parameter appearing in condition Sβ , holding of which is implied by the condition T4 . Two cases, where the vector parameter (γ, β) takes the value (0, ∞) or (∞, 0), are excluded, since, according to the Remarks 5.2 and 5.3, Theorems 5.1 and 5.2 do not work in these cases. Also worth mentioning is the case where there exists a limiting alternating regenerative process. This is the case, where it is assumed that there exist some limiting stochastic triplets ξ¯0,i,n = ξ0,n (t), t ≥ 0 , κ0,i,n , η0,i,n , i = 1, 2, n = 1, 2, . . . possessing the properties (F)–(J) and the corresponding alternating limiting regenerative process is defined using the above stochastic triplets and relations similar to relations (3.1)–(3.3), ξ0 (t) = ξ0,η0, u0, n−1,n (t − ζ0,n−1 ) and η0 (t) = η0,n−1, for t ∈ [ζ0,n−1, ζ0,n ), n = 1, 2, . . . ,

(5.56)

ζ0,n = κ0,1 + · · · + κ0,n, n = 1, 2, . . . , ζ0,0 = 0.

(5.57)

where, In this case, the transition characteristics of the limiting alternating regenerative process (ξ0 (t), η0 (t)) are connected with the transition characteristics of the compressed in time alternating regenerative processes (ξε,uε (t), ηε,uε (t)) by the conˆ 2, X ˆ γ and the relations (5.42)–(5.54), and, thus, they are given by the ditions Pˆ 2 –R following relations: (γ)

Q0,i j (t) = Q0,i j (t), t ≥ 0, i, j ∈ X, (γ)

p0,i j = Q0,i j (∞), i, j ∈ X, (γ)

e0,i j = e0,i j , i, j ∈ X, (γ)

q0,i (t, A) = q0,i (t, A), t ≥ 0, A ∈ BZ, i ∈ X.

(5.58)

Chapter 6

Super-Long and Long Time Ergodic Theorems for Singularly Perturbed Alternating Regenerative Processes

In this chapter, we present super-long and long time ergodic theorems for perturbed alternating regenerative processes modulated by singularly perturbed two-state semiMarkov processes (shortly referred as singularly perturbed alternating regenerative processes). This is the case, where both regime switching probabilities for perturbed modulating semi-Markov processes are positive but converge to zero, as ε → 0. This chapter includes three sections. In Sect. 6.1, we consider singularly perturbed alternating regenerative processes and find a form of aggregation for regeneration times suitable for ergodic analysis of such processes. In Sect. 6.2, we study super-long time ergodic asymptotics for singularly perturbed alternating regenerative processes and present it in Theorem 6.1. In Sect. 6.3, we study long-time ergodic asymptotics for singularly perturbed alternating regenerative processes and present it in Theorem 6.2.

6.1 Singularly Perturbed Alternating Regenerative Processes and Aggregation of Regeneration Times In this section, we introduce the model of singularly perturbed alternating regenerative processes and find a form of aggregation for regeneration times appropriate for the ergodic analysis of such processes.

6.1.1 Singularly Perturbed Alternating Regenerative Processes These are alternating regenerative processes with a singular perturbation model, where, in addition to O2 –R2 , the following condition is satisfied: U1 : 0 < pε,12 → p0,12 = 0 as ε → 0 and 0 < pε,21 → p0,21 = 0 as ε → 0. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0_6

107

108

6 Super-long and long time ergodic theorems for singularly perturbed ARP

Here, we should also assume that the probabilities pε,12 and pε,21 are asymptotically comparable in the sense that the condition Sβ holds for some β ∈ [0, ∞], i.e., the following relation holds: βε =

pε,12 → β ∈ [0, ∞] as ε → 0. pε,21

(6.1)

In the case of singularly perturbed alternating regenerative processes, the limiting embedded Markov chain η0,n has the matrix of transition probabilities p0,i j  = I(i = j), and, thus, the Markov chain η0,n is not ergodic. In what follows in this section, we will assume that (ξε (t), ηε (t)) is, for every ε ∈ (0, 1], an alternating regenerative process with regeneration times ζε,n . We also assume that the conditions O2 –R2 , U1 , and Sβ are satisfied for the alternating regenerative processes (ξε (t), ηε (t)). It is useful noting that, in the case where the condition U1 is satisfied, the condition P2 can be simplified and weakened. Indeed, in this case, the condition P2 (a) is automatically satisfied with the corresponding limiting probabilities p0,i j = I(i = j), i, j ∈ X. Thus, the sets Y0,i = {i}, i ∈ X. The limiting distribution functions Q0,ii (·) = F0,ii (·) = F0,i (·), i ∈ X and Q0,i j (·) = I(· ≥ 0), j  i, i ∈ X, in the condition P2 (b). The condition P2 (c) reduces to the assumption that the distribution functions F0,i (·), i ∈ X are non-arithmetic. Moreover, the non-arithmetic assumption can be replaced by the assumption that the distribution functions F0,i (·), i ∈ X are weakly non-arithmetic. Thus, according to the above remarks, the condition P2 can, in this case where the condition U1 is satisfied, be replaced by the following weaker condition: P¯ 2 : Q ε,ii (·) ⇒ Q0,ii (·) as ε → 0, for i ∈ X, where Q0,ii (·) = F0,i (·) is a weakly non-arithmetic distribution function, for i ∈ X. Also, according to the above remarks, the first moments e0,ii = e0,i, i ∈ X, while e0,i j = 0, j  i, i ∈ X, in the condition Q2 .

6.1.2 Embedded Regenerative Processes of the Second Type 6.1.2.1 Renewal Equations for Distributions of Embedded Regenerative Processes of the Second Type. Let us recall the function, −1 vε = p−1 ε,12 + pε,21, ε ∈ (0, 1].

(6.2)

The condition U1 , obviously, implies that, 0 < vε → v0 = ∞ as ε → 0.

(6.3)

Let us consider, for every ε ∈ (0, 1], a compressed in time embedded regenerative processes of the second type ξˇε,vε (t) = (ξε,vε (t), ηε,vε (t)) = ξˇε (tvε ) =

6.1 Singularly perturbed ARP and aggregation of regeneration times

109

(ξε (tvε ), ηε (tvε )) with compressed in time aggregated regeneration times ζˇε,vε ,n = vε−1 ζˇε,n . The condition U1 implies that Pi { ζˇε,vε ,n < ∞} = 1, for n = 0, 1, . . . , i ∈ X. The process ξˇε,vε (t) has the phase space Z = Z × X. The σ-algebra of measurable sets BZ includes sets of the form, B = (A1 × {1}) ∪ (A2 × {2}), A1, A2 ∈ BZ . Let us consider the probabilities, Pˇε,vε ,i (t, B) = Pi { ξˇε,vε (t) ∈ B} = Pi {(ξε (tvε ), ηε (tvε ) ∈ B} = Pi {(ξε (tvε ), ηε (tvε )) ∈ A1 × {1}} + Pi {(ξε (tvε ), ηε (tvε )) ∈ A2 × {2}} = Pε,i1 (tvε, A1 ) + Pε,i2 (tvε, A2 ), t ∈ R+, B ∈ BZ .

(6.4)

The idea behind considering compressed in time embedded regenerative processes ζˇε,vε (t) with regenerative times ζˇε,vε ,n is to show that the conditions O1 –R1 are satisfied for these processes, then to apply the ergodic Theorems 2.1–2.3 to these processes, and, in this way, to obtain the corresponding ergodic relations for the probabilities Pˇε,vε ,i (tε, B) and Pε,i j (tε vε, A) as 0 ≤ tε → ∞ as ε → 0. The function Pˇε,vε ,i (·, B) is, for every B ∈ BZˆ , the unique in the class L solution for the following renewal equation: ∫ t ˇ (6.5) Pˇε,vε ,i (t − s, B)Fˇε,vε ,i (ds), t ≥ 0, Pε,vε ,i (t, B) = qˇε,vε ,i (t, B) + 0

where, for t ∈ R+ ,

Fˇε,vε ,i (t) = Pi { ζˇε,vε ,1 ≤ t},

(6.6)

qˇε,vε ,i (t, B) = Pi { ξˇε,vε (t) ∈ B, ζˇε,vε ,1 > t} = Pi {(ξε,vε (t), ηε,vε (t)) ∈ B, ζˇε,vε ,1 > t}.

(6.7)

ˆ ˆ. and, for t ∈ R+, B ∈ B Z

The function qˇε,vε ,i (t, B), t ∈ R+, B ∈ BZˆ belongs to the class P[BZˆ ]. Moreover, it is consistent, for ε ∈ (0, 1], with the tail probability function 1 − Fˇε,vε ,i (·), i.e., the following relation holds: ˆ = 1 − Fˇε,vε ,i (t), for t ∈ R+ . qˇε,vε ,i (t, Z)

(6.8)

˜ ν˜ε,n−1 ], n = 1, 2, . . ., Consider again the random variables ν˜ε,0 = 0, ν˜ε,n = θ[ which are successive moments of state change by the Markov chain ηε,n and the random variables ζ˜ε,n = ζε,ν˜ ε, n , n = 0, 1, . . ., which are successive moments of state change by the semi-Markov process ηε (t). The random variables ζ˜ε,vε ,n = ve−1 ζ˜ε,n, n = 0, 1, . . . are successive moments of state change by the semi-Markov process ηε,vε (t). The random variables ζˇε,vε ,n and ζ˜ε,vε ,n are connected by the following relation:

110

6 Super-long and long time ergodic theorems for singularly perturbed ARP

ζˇε,vε ,n = ζ˜ε,vε ,2n, n = 0, 1, . . . .

(6.9)

That is why, for t ≥ 0, Fˇε,vε ,i (t) = Pi { ζˇε,vε ,1 ≤ t} = Pi { ζ˜ε,vε ,2 ≤ t},

(6.10)

ˆ ˆ. and, for t ≥ 0, B ∈ B Z qˇε,vε ,i (t, B) = Pi { ξˇε,vε (t) ∈ B, ζˇε,vε ,1 > t} = Pi {(ξε,vε (t), ηε,vε (t)) ∈ B, ζ˜ε,vε ,2 > t}.

(6.11)

Let us introduce, for i ∈ X and ε ∈ (0, 1], the distribution function, F˜ε,vε ,i (t) = P{ ζ˜ε,vε ,1 ≤ t}, t ≥ 0

(6.12)

q˜ε,vε ,i (t, A) = Pi {ξε,vε (t) ∈ A, ζ˜ε,vε ,1 ≥ t}, t ≥ 0, A ∈ BZ .

(6.13)

and the function,

The following relation takes place for the random variables ζ˜ε,vε ,1 and ζ˜ε,vε ,2 − ζ˜ε,vε ,1 , for i, j ∈ X, j  i: Pi { ζ˜ε,vε ,1 ≤ t , ζ˜ε,vε ,2 − ζ˜ε,vε ,1 ≤ t  }

= Pi { ζ˜ε,vε ,1 ≤ t  }P j { ζ˜ε,vε ,2 − ζ˜ε,vε ,1 ≤ t  } = F˜ε,vε ,i (t )F˜ε,vε , j (t ), t , t  ≥ 0.

(6.14)

The relation (6.14) implies that the random variables ζ˜ε,vε ,1 and ζ˜ε,vε ,2 − ζ˜ε,vε ,1 are independent, for every i, j ∈ X, j  i, It also follows from the relation (6.14) that the tail probability function 1− Fˇε,vε ,i (·) has the following form, for i ∈ X, j  i: 1 − Fˇε,vε ,i (t) = 1 − F˜ε,vε ,i (t) ∗ F˜ε,vε , j (t), t ≥ 0.

(6.15)

Obviously, if ηε,vε (0) = i, then ηε,vε (t) = i for t ∈ [0, ζ˜ε,vε ,1 ), and ηε,vε (t) = j  i, for t ∈ [ζ˜ε,vε ,1, ζˇε,vε ,1 ). Using this remark, we get the following relation, for t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}), A1, A2 ∈ BZ, i ∈ X: qˇε,vε ,i (t, B) = Pi {ξε,vε (t) ∈ A1, ηε,vε (t) = i, ζ˜ε,vε ,1 > t} + Pi { ζ˜ε,vε ,1 ≤ t, ξε,vε (t) ∈ A2, ηε,vε (t) = j, ζ˜ε,vε ,2 > t} ∫ t (6.16) = q˜ε,vε ,i (t, A1 ) + F˜ε,vε ,i (ds)q˜ε,vε , j (t − s, A2 ). 0

It is useful noting that Z = (Z × {1}) ∪ (Z × {2}) = Z × X. In this case, the relation (6.16) takes the following consistent with the relation (6.15) form, for t ∈ R+, j  i:

6.1 Singularly perturbed ARP and aggregation of regeneration times

 = q˜ε,vε ,i (t, Z) + qˇε,vε ,i (t, Z)

∫

t

F˜ε,vε ,i j (ds)q˜ε,vε ,i (t − s, Z)

0

∫

= 1 − F˜ε,vε ,i (t) +

111

t

0

F˜ε,vε ,i (ds)(1 − F˜ε,vε , j (t − s))

= 1 − F˜ε,vε ,i (t) ∗ F˜ε,vε , j (t) = 1 − Fˇε,vε ,i (t).

(6.17)

In Sect. 6.1.2.4, we shall get the following weak convergence relation, for i ∈ X: F˜ε,vε ,i (·) ⇒ F˜0,i (·) as ε → 0,

(6.18)

where F˜0,i (·) is a proper distribution function. Moreover, we shall find the explicit form for the distribution functions F˜0,i (·), i ∈ X. We also shall prove that the following relation takes place for the first moments of the above distribution functions: ∫ ∞ e˜ε,vε ,i = t F˜ε,vε ,i (dt) 0 ∫ ∞ → e˜0,i = t F˜0,i (dt) < ∞ as ε → 0. (6.19) 0

Moreover, we shall find the explicit form for the expectations e˜0,i, i ∈ X. Finally, we shall prove that the condition R2 is satisfied for the functions q˜ε,vε ,i (t, A), t ∈ R+, A ∈ BZ , i ∈ X, that is, there exist functions q˜0,i (t, A), t ≥ 0, A ∈ BZ, i ∈ X which belongs to the class P[BZ ], a class of sets Γ ⊆ BZ , and Borel sets U[q˜ ·,v·,i (·, A)], A ∈ Γ, i ∈ X such that: (a) the function q˜0,i (t, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − F˜0,i (t), t ∈ R+ , for i ∈ X; (b) the funcus tions q˜ε,vε ,i (·, A) −→ q˜0,i (·, A) as ε → 0, for points s ∈ U[q˜ ·,v·,i (·, A)], A ∈ Γ, i ∈ X; (c) m(U¯ [q˜ ·,v·,i (·, A)]) = 0, for A ∈ Γ, i ∈ X; (d) the function q˜0,i (·, A) is continuous almost everywhere with respect to the Lebesgue measure m(·) on B+ , for A ∈ Γ, i ∈ X. We also find the explicit form for the functions q˜0,i (·, A), A ∈ Γ, i = 1, 2 and show that the role of the class Γ can be played by the class Γ appearing in the condition R2 . The relations (6.15) and (6.16) indicate that the corresponding limiting distribution function Fˇ0,i (t) and the function qˇ0,i (t, B) (which should appear for the embedded regenerative processes ξˇε,vε (t) in the conditions P1 and R1 ) can be defined by the following relations similar to (6.15) and (6.16), that is, for t ∈ R+, j  i: 1 − Fˇ0,i (t) = 1 − F˜0,i (t) ∗ F˜0,v0, j (t), and, for t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}), A1, A2 ∈ BZ, j  i, ∫ t qˇ0,i (t, B) = q˜0,i (t, A1 ) + F˜0,i (ds)q˜0, j (t − s, A j ). 0

(6.20)

(6.21)

112

6 Super-long and long time ergodic theorems for singularly perturbed ARP

The functions q˜0, j (t, A), t ∈ R+, A ∈ BZ, j = 1, 2 belong to the class P[BZ ]. This implies that the functions qˇ0,i (t, B), t ∈ R+, B ∈ BZ , i = 1, 2 given by the relation (6.21) also belong to the class P[BZ ]. The function q˜0,i (·, A), t ∈ R+, A ∈ BZ, is consistent with the tail probability function 1 − F˜0,i (·), for i = 1, 2. This implies that the function qˇ0,i (t, B), t ∈ R+, B ∈ BZ given by the relation (6.21) is consistent with the tail probability function 1 − Fˇ0,i (·), for i = 1, 2. Note also the case where the existence of a limiting alternating regenerating process (ξ0 (t), η0 (t)) with regeneration times ζ0,n is assumed. In this case, it should be also assumed that the distribution functions F0,i (t), the functions q0,i (t, A), and other limiting quantities appearing in the conditions O2 , P¯ 2 , Q2 , and R2 are defined via the corresponding limiting alternating regenerative process. The above-described construction of embedded regenerative processes of the second type is used below for obtaining ergodic theorems for singularly and supersingularly perturbed alternating regenerative processes. These results are presented in Theorems 6.1, 6.2, and 7.1–7.4. 6.1.2.2 Compressed in Time Embedded Regenerative Processes of the Second Type. In this case, it is assumed that, for every ε ∈ (0, 1], conditions O2 , P¯ 2 , Q2 , and R2 are satisfied for the compressed in times alternating regenerative processes (ξε,uε (t), ηε,uε (t)) = (ξε (tuε ), ηε (tuε )) with regeneration times ζε,uε ,n = u−1 ε ζε,n , where uε, ε ∈ (0, 1] is a positive function serving as a time compression factor. In this case, the quantities pε,i j , Q ε,i j (t), Fε,i (t), eε,i , and qε,i (t, A) defined for the alternating regenerative processes in Sect. 3.1.2.1 should be replaced by the quantities pε,uε ,i j = pε,i j , Q ε,uε ,i j (t) = Q ε,i j (tuε ), Fε,uε ,i (t) = Fε,i (tuε ), eε,uε ,i = u−1 ε eε,i , and qε,uε ,i (t, A) = qε,i (tuε, A) in conditions O2 –R2 , while the corresponding limiting quantities p0,u0,i j , Q0,i j (t), F0,i (t), e0,i , and q0,i (t, A) do not change in these conditions. The corresponding embedded regenerative process of the first type constructed with the use of procedure described in Sect. 3.3.2.2 is also compressed in time. It is the process ξˇε,vε uε (t) = (ξε (tvε uε ), ηε (tvε uε )) with regeneration times ζˇε,vε uε ,n = −1 ˇ u−1 ε ζε,vε ,n = (vε uε ) ζε,n . The relation (4.17) takes in this case the following form: Pˇε,vε uε ,i (t, B) = Pi { ξˇε,vε uε (t) ∈ B} = Pi {(ξε (tvε uε ), ηε (tvε uε ) ∈ B} = Pi {(ξε (tvε uε ), ηε (tvε uε )) ∈ A1 × {1}} + Pi {(ξε (tvε uε ), ηε (tvε uε )) ∈ A2 × {2}} = Pε,i1 (tvε uε, A1 ) + Pε,i2 (tvε uε, A2 ), t ∈ R+, B ∈ BZ .

(6.22)

As above, the idea behind considering embedded regenerative processes ζˇε,vε uε (t) with regenerative times ζˇε,vε uε ,n is to show that the conditions O1 –R1 are satisfied for these processes. Then one should apply the ergodic Theorems 2.1–2.3 to these processes and, in this way, obtain the corresponding ergodic relations for the probabilities Pˇε,vε uε ,i (tε, B) and Pε,i j (tε vε uε, A) as 0 ≤ tε → ∞ as ε → 0.

6.1 Singularly perturbed ARP and aggregation of regeneration times

113

The described above construction of compressed in time embedded regenerative processes of the second type for singularly perturbed alternating regenerative processes is described below, in Sect. 8.2. This makes it possible to generalise ergodic Theorems 6.1, 6.2 and 7.1–7.4 to the model of singularly perturbed alternating regenerative processes compressed in time. 6.1.2.3 Embedded Regenerative Processes of the First Type. Ergodic theorems 2.1–2.3 for perturbed regenerative processes require weak convergence of distributions for regeneration times and convergence of expectations for regeneration times to the first moment of the corresponding limiting distribution for regeneration times. As follows from the relations (6.18) and (6.19), the indicated simultaneous convergence takes place for the compressed in time embedded regenerative processes of the second type ξˇε,vε (t) = (ξε (tvε ), ηε (tvε )) with the regeneration times ζˇε,vε ,n = vε−1 ζε,n . However, aforementioned simultaneous convergence may not be the case for the embedded regenerative processes of the first type ξˆε (t) = (ξε (t), ηε (t)) with the regeneration times ζˆε,n . Let, for example, the initial state ηε (0) = 1. In this case, the distribution function of the regeneration time, Fˆε,11 (t) = P1 { ζˆε,1 ≤ t}, t ≥ 0.

(6.23)

The relation (4.9) implies that, φˆε,11 (s) = ψε,11 (s) + ψε,12 (s)

ψε,21 (s) , s ≥ 0, 1 − ψε,22 (s)

(6.24)

where, for i, j ∈ X, ∫ ψε,i j (s) =

0

∞

e−st Q ε,i j (dt), s ≥ 0.

(6.25)

The conditions P¯ 2 and U1 imply that, for s ≥ 0, j  i, ψε,i j (s) ≤ pε,i j → 0 as ε → 0,

(6.26)

ψε,ii (s) → ψ0,ii (s) = φ0,ii (s) as ε → 0.

(6.27)

while, for s ≥ 0, i ∈ X,

The relations (6.24)–(6.27) imply that, for s ≥ 0, φˆε,11 (s) → φˆ0,11 (s) = ψ0,11 (s) = φ0,11 (s) as ε → 0,

(6.28)

and, thus, the distributions of regeneration times, Fˆε,11 (·) ⇒ Fˆ0,11 (·) = Q0,11 (·) as ε → 0.

(6.29)

114

6 Super-long and long time ergodic theorems for singularly perturbed ARP

Note that the limiting distribution has the first moment, ∫ ∞ eˆ0,11 = e0,11 = tQ0,11 (dt) < ∞.

(6.30)

0

At the same time, the relation (4.57) imply that, in this case, eˆε,11 =

eε,1 pε,21 + eε,2 pε,12 , pε,21

(6.31)

where, for i ∈ X and ε ∈ [0, 1], eε,i =

∫ j ∈X

∞

0

tQ ε,i j (dt).

(6.32)

The conditions U1 , Sβ , P¯ 2 , Q2 , and the relation (4.57) imply that, in this case,

If β > 0, then,

eˆε,11 → e0,1 + e0,2 β as ε → 0.

(6.33)

eˆ0,11 = e0,11  e0,1 + e0,2 β.

(6.34)

This makes it impossible to use Theorems 2.1–2.3, which require the convergence of the expectations for the regeneration times to the first moment of the corresponding limiting distribution for the regeneration times. 6.1.2.4 Simultaneous Convergence of Distribution Functions and Their First Moments for Regeneration Times of Embedded Regenerative Processes of the Second Type. The aforementioned simultaneous convergence takes place for the compressed in time embedded regenerative processes of the second type ξˇε,vε (t) = (ξε (tvε ), ηε (tvε )) with the regeneration times ζˇε,vε ,n = vε−1 ζˇε,n . Let the initial state ηε (0) = i ∈ X. In this case, the distribution function of the regeneration time ζˇε,vε ,1 takes, according to the relation (6.14), the following form, for j  i: Fˇε,vε ,i (t) = Pi { ζˇε,vε ,1 ≤ t} = Pi { ζ˜ε,vε ,2 ≤ t} = F˜ε,vε ,i (t) ∗ F˜ε,vε , j (t), t ≥ 0.

(6.35)

Let ci,β be a function defined on the set [0, ∞]×X and taking values in the interval [1, ∞],  1 + β for β ∈ [0, ∞], i = 1, (6.36) ci,β = 1 + β−1 for β ∈ [0, ∞], i = 2. The following useful lemmas take place. Lemma 6.1 Let the conditions O2 , P¯ 2 , Q2 , U1 , and Sβ be satisfied. Then, the following relation takes place, for i ∈ X:

6.1 Singularly perturbed ARP and aggregation of regeneration times

F˜ε,vε ,i (·) ⇒ F˜0,i (·) = P{

e0,i ζ ≤ ·} as ε → 0, ci,β

115

(6.37)

where ζ is a random variable exponentially distributed with parameter 1. Lemma 6.2 Let the conditions O2 , P¯ 2 , Q2 , U1 , and Sβ be satisfied. Then, the following relation takes place, for i ∈ X: Fˇε,vε ,i (·) ⇒ Fˇ0,i (·) = Fˇ0 (·) = P{

e0,1 e0,2 ζ1 + ζ2 ≤ ·} as ε → 0, c1,β c2,β

(6.38)

where ζ1 and ζ2 are independent random variables exponentially distributed with parameter 1. Proof It is convenient to use the definition of an alternating regenerative process (ξε (t), ηε (t)) with regeneration times ζε,n in terms of stochastic triplets ξ¯ε,i,n = ξε,i,n (t), t ≥ 0 , κε,i,n, ηε,i,n , i = 1, 2, n = 1, 2, . . ., which is given by the relations (3.1)–(3.3). In the case, where the initial state ηε (0) = i ∈ X, the random variable ζ˜ε,vε ,1 can be represented in the form of the following random sum: ζ˜ε,vε ,1 = vε−1

θ

ε [0]

κε,i,n,

(6.39)

n=1

where: (a) the random index θ ε [0] = min(n ≥ 1 : ηε,i,n = j) (here, j  i) has the geometric distribution with parameter pε,i j , i.e., it takes value n with probability n−1 p pε,ii ε,i j , for n = 1, 2, . . ., (b) κε,i,n, n = 1, 2, . . . are i.i.d. random variables with the distribution function Fε,i (t) = P{κε,i,1 ≤ t} = Q ε,i1 (t) + Q ε,i2 (t), t ≥ 0. It should be noted that the random index θ ε [0] and the sequence of random variables κε,i,n, n = 1, 2, . . . are dependent. More precisely, (c) θ ε [0] = min(n ≥ 1 : ηε,i,n = j), (d) (κε,1,n, ηε,1,n ), n = 1, 2, . . . are i.i.d. random vectors, with the joint distribution P{κε,i,1 ≤ t, ηε,i,1 = j} = Q ε,i j (t), t ≥ 0, j ∈ X. The condition U1 implies that, for j  i, d

pε,i j θ ε [0] −→ ζ as ε → 0,

(6.40)

where ζ is a random variable exponentially distributed, with parameter 1. The conditions P¯ 2 and Q2 imply that, Fε,i (·) = P{κε,i,1 ≤ ·} ⇒ F0,i (·) as ε → 0, and

∫ eε,i = E κε,i,1 =

0

∞

sFε,i (ds)

(6.41)

116

6 Super-long and long time ergodic theorems for singularly perturbed ARP

∫ → e0,i =

∞

sF0,i (ds) < ∞ as ε → 0.

(6.42)

0

The relations (6.41) and (6.42) imply that, for any function 0 < nε → ∞ as ε → 0, [n ε]

d n−1 κε,i,n −→ e0,i as ε → 0. (6.43) ε k=1

Indeed, let 0 < sk → ∞ as k → ∞ be a sequence of continuity points for the distribution function F0,i (s). The relations (6.41) and (6.42) obviously imply that, for any t > 0, ∫ ∞ ∫ ∞ lim sFε,i (ds) ≤ lim sFε,i (ds) ε→0 tnε ε→0 sk ∫ sk sFε,i (ds)) = lim (eε,i − ε→0 0 ∫ sk sF0,i (ds) → 0 as k → ∞, (6.44) = e0,i − 0

and, thus, the following relation holds, for any t > 0: ∫ ∞ lim sFε,i (ds) = 0. ε→0

(6.45)

tnε

The relation (6.45) implies that, for any t > 0, nε P{n−1 ε κε,i,1 > t} = nε (1 − Fε,i (tnε )) ∫ ∞ ≤ t −1 sFε,i (ds) → 0 as ε → 0.

(6.46)

tnε

Also, the relations (6.44) and (6.45) imply that, for any t > 0, −1 nε En−1 ε κε,i,1 I(nε κε,i,1 ≤ t) ∫ tnε sdFε,i (ds) → e0,i as ε → 0. =

(6.47)

0

The relations (6.46) and (6.47) imply, by the criterion of central convergence (see, for example, Loève (1977)), that the relation (6.43) holds. As was mentioned above, the random index θ ε [0] and the random variables κε,i,n, n = 1, 2, . . . can be dependent. Nevertheless, since the limit in relation (6.43) is non-random, the relations (6.40) and (6.43) imply, by Theorem B.4, that, for j  i, (pε,i j θ ε [0],

−1 n ≤t pε, ij

d

κε,i,n, t ≥ 0 −→ (ζ, te0,i ), t ≥ 0 as ε → 0.

(6.48)

6.1 Singularly perturbed ARP and aggregation of regeneration times

117

The relation (6.48) implies, by Theorem B.1 that the following relation holds: d

pε,i j ζ˜ε,1 −→ e0,i ζ as ε → 0.

(6.49)

The condition Sβ implies that pε,i j vε → ci,β as ε → 0 and, thus, the relation (6.49) implies that the following relation holds, for i ∈ X: e0,i F˜ε,vε ,i (·) ⇒ F˜0,i (·) = P{ ζ ≤ ·} as ε → 0. ci,β

(6.50)

Relations (6.35) and (6.50) imply that, for i ∈ X, j  i, Fˇε,vε ,i (·) = F˜ε,ve,i (t) ∗ F˜ε,vε , j (t) ⇒ Fˇ0,i (·) as ε → 0,

(6.51)

where, Fˇ0,i (t) = F˜0,i (t) ∗ F˜0, j (t) e0,1 e0,2 = Fˇ0 (t) = P{ ζ1 + ζ2 ≤ t}, t ≥ 0. c1,β c2,β

(6.52) 

The proof is complete. The following useful lemmas take place.

Lemma 6.3 Let the conditions O2 , P¯ 2 , Q2 , U1 , and Sβ be satisfied. Then, the following relation takes place, for i ∈ X: ∫ ∞ t F˜ε,vε ,i (dt) e˜ε,vε ,i = 0

∫

→ e˜0,i = 0

∞

t F˜0,i (dt) =

e0,i as ε → 0. ci,β

(6.53)

and Lemma 6.4 Let the conditions O2 , P¯ 2 , Q2 , U1 , and Sβ be satisfied. Then, the following relation takes place, for i ∈ X: ∫ ∞ ∫ ∞ t Fˇε,vε ,i (dt) → eˇ0,i = t Fˇ0,i (dt) eˇε,vε ,i = 0 0 ∫ ∞ e0,1 e0,2 = t Fˇ0 (dt) = eˇ0 = + as ε → 0. (6.54) c c2,β 1,β 0 Proof The representation (6.39) for the random variable ζ˜ε,vε ,1 as a random sum implies that, for j  i,

118

6 Super-long and long time ergodic theorems for singularly perturbed ARP

e˜ε,vε ,i = vε−1 E

θ

ε [0]

κε,i,n

n=1

= vε−1 E

∞

κε,i,n I(θ ε [0] > n − 1)

n=1

= vε−1 E

∞

κε,i,n I(ηε,i,k = 1, 1 ≤ k ≤ n − 1)

n=1

= vε−1

∞

E κε,i,n EI(ηε,i,k = 1, 1 ≤ k ≤ n − 1)

n=1

= vε−1

∞

n−1 eε,i pε,ii =

n=1

eε,i . vε pε,i j

(6.55)

The relations (6.35) and (6.55) imply that the following relation holds, for i ∈ X: eˇε,vε ,i = e˜ε,vε ,1 + e˜ε,vε ,2 1 1 + eε,2 vε pε,12 vε pε,21 e0,1 e0,2 → eˇ0,i = eˇ0 = + as ε → 0. c1,β c2,β = eε,1

The proof is complete.

(6.56) 

The relations (6.51) and (6.56) show that the time compression factor vε has been chosen correctly. The convergence of the expectations of the regeneration times to the first moment of the corresponding limiting distribution for the regeneration times takes place for the embedded regenerative processes ξˇε (t). This makes it possible to use Theorems 2.1–2.3 for obtaining ergodic theorems for singularly perturbed alternating regenerative processes. Remark 6.1 From the proof of Lemmas 6.1–6.4 it is easy to see that the condition P¯ 2 can be weakened by omitting the assumption that the limiting distribution functions F0,ii (·), i = 1, 2 appearing in this condition are weakly non-arithmetic (the condition U1 implies that the set Y0,i = {i}, for i = 1, 2). The proof also admits cases, where one or both of the limiting distributions F0,ii (·) coincide with the distribution function I(· ≥ 0), and, thus, one or both moments e0,i = 0. The distribution function F˜0,1 (·) is not concentrated at zero if and only if e0,1 > 0, β ∈ [0, ∞). The distribution function F˜0,2 (·) is not concentrated at zero if and only if e0,2 > 0, β ∈ (0, ∞]. The distribution function Fˇ0 (·) is not concentrated at zero if and only if e0,1, e0,2 > 0, or e0,1 > 0, e0,2 = 0, β ∈ [0, ∞), or e0,1 = 0, e0,2 > 0, β ∈ (0, ∞].

6.1 Singularly perturbed ARP and aggregation of regeneration times

119

Remark 6.2 Obviously, any of the limiting distribution functions, Fˇ0,1 (·), Fˇ0,2 (·), and Fˇ0 (·), is weakly non-arithmetic if it is not concentrated at zero. Thus, the assumption that F0,ii (·), i = 1, 2 are weakly non-arithmetic can be replaced in the condition P¯ 2 by the corresponding assumptions (given in Remark 6.1), which guarantee that the above distribution functions are not concentrated at zero. 6.1.2.5 Condition R2 for functions q˜ε,vε ,i (t, A). These functions are given by the following relation, for i ∈ X and ε ∈ (0, 1]: q˜ε,vε ,i (t, A) = Pi {ξε,vε (t) ∈ A, ζ˜ε,vε ,1 ≥ t}, t ≥ 0, A ∈ BZ .

(6.57)

Let us introduce, for ε ∈ (0, 1], the stochastic triplets, ξ¯ε,i,n = ξε,i,n (t), t ≥ 0 , κε,i,n, με,i,n , i = 1, 2, n = 1, 2, . . . ,

(6.58)

where the binary random variables με,i,n are defined by the following relation: με,i,n = κε,i,n I(ηε,i,n = i), i = 1, 2, n = 1, 2, . . . .

(6.59)

Also, let us and define, for i ∈ X and ε ∈ [0, 1], the regenerative process, ξε,i (t) = ξε,i,n (t − ζε,i,n−1 ), for t ∈ [ζε,i,n−1, ζε,i,n ), n = 1, 2, . . . ,

(6.60)

with the regeneration times, ζε,i,n = κε,i,1 + · · · + κε,i,n, n = 1, 2, . . . , ζε,i,0 = 0,

(6.61)

and the regenerative lifetime, με,i,+ = ζε,i,νε, i ,

(6.62)

where, for j  i, νε,i = min(n ≥ 1 : με,i,n < κε,i,n ) = min(n ≥ 1 : ηε,i,n = j).

(6.63)

Let us also consider the following probabilities, for t ≥ 0, A ∈ Γ and i ∈ X: Pε,i,+ (t, A) = P{ξε,i (t) ∈ A, με,i,+ > t}.

(6.64)

In this case, the corresponding distribution functions of regeneration times, F¯ε,i (u) = P{ζε,i,1 ≤ u} = P{κε,i,1 ≤ u}, u ≥ 0, and

(6.65)

120

6 Super-long and long time ergodic theorems for singularly perturbed ARP

Fε,i,+ (u) = P{ζε,i,1 ≤ u, με,i,+ ≥ ζε,i,1 } = P{κε,i,1 ≤ u, με,i,1 ≥ κε,i,1 } = P{κε,i,1 ≤ u, ηε,i,1 = i}, u ≥ 0.

(6.66)

The corresponding renewal equation for the probabilities Pε,i,+ (t, A), t ≥ 0 takes the following form: ∫ t Pε,i,+ (t, A) = qε,i (t, A) + Pε,i,+ (t − s, A)Fε,i,+ (ds), t ≥ 0, (6.67) 0

where qε,i (t, A) = P{ξε,i,1 (t) ∈ A, ζε,i,1 ∧ με,i,+ > t} = P{ξε,i,1 (t) ∈ A, ζε,i,1 > t}, t ≥ 0.

(6.68)

Here, it is used that P{με,i,+ ≥ ζε,i,1 } = 1 according to the relations (6.62) and (6.63). The corresponding stopping probability takes the following form for j  i: qε,i,+ = P{με,i,1 < κε,i,1 } = Pi {ηε,i,1 = j} = pε,i j .

(6.69)

The conditions U1 imply that, for i ∈ X, 0 < qε,i,+ → 0 as ε → 0.

(6.70)

In addition, the corresponding expectations of inter-regeneration times, e¯ε,i = E κε,i,1 = eε,i1 + eε,i2,

(6.71)

and eε,i,+ = E κε,i,1 I(με,i,1 ≥ κε,i,1 ) = E κε,i,1 I(ηε,i,1 = i) = eε,i1 .

(6.72)

The following relation plays the key role in what follows. It holds, for t ≥ 0, A ∈ BZ, i ∈ X, q˜ε,vε ,i (t, A) = Pi {ξε (tvε ) ∈ A, ζ˜ε,1 > tvε } = P{ξε,i (tvε ) ∈ A, με,i,+ > tvε } = Pε,i,+ (tvε, A).

(6.73)

Note that c1,β = ∞, if β = ∞, and c2,β = ∞, if β = 0. Lemma 6.5 Let the conditions O2 , P¯ 2 , Q2 , R2 , U1 , and Sβ (for some β ∈ [0, ∞]) be satisfied. Then the conditions O1 –R1 are satisfied for the regenerative processes ξε,i (t), t ≥ 0, with the regenerative times ζε,i,n, n = 1, 2, . . ., and the regenerative

6.1 Singularly perturbed ARP and aggregation of regeneration times

121

lifetimes με,i,+ , for i ∈ X, and the following relation takes place for any A ∈ Γ and 0 ≤ sε → s ∈ (0, ∞) as ε → 0: Pε,i,+ (sε vε, A) = q˜ε,vε ,i (sε, A) → q˜0,i (s, A) = e−sci, β /e0, i π0,i (A) as ε → 0,

(6.74)

Proof The condition O2 obviously implies that the condition O1 (a) is satisfied for the regenerative processes ξε,i (t), for i ∈ X. The relation (6.70) implies that the condition O1 (b) is also satisfied. The conditions P¯ 2 and U1 obviously imply that, F¯ε,i (·) ⇒ F¯0,i (·) as ε → 0,

(6.75)

where the limiting distribution F¯0,i (·) = Q0,ii (·) = F0,ii (·), for i ∈ X, since the limiting probabilities p0,ii = 1, i ∈ X. Moreover, according to the condition P¯ 2 , the distribution function F0,i,+ (·) is non-arithmetic, for i ∈ X. Also, the relations (6.65) and (6.66) imply that the following inequality takes place: for t ≥ 0, i ∈ X, j  i and ε ∈ (0, 1], 0 ≤ F¯ε,i (t) − Fε,i,+ (t) ≤ P{ηε,i,1 = j} = pε,i j .

(6.76)

The relations (6.75) and (6.76) imply that, for i ∈ X, Fε,i,+ (·) ⇒ F0,i,+ (·) = F¯0,i (·) as ε → 0.

(6.77)

Therefore, the condition P1 is satisfied for the regenerative processes ξε,i (t), for i ∈ X. The conditions Q2 and U1 obviously imply that the condition Q1 is satisfied, for i ∈ X, that is, ∫ ∞ e¯ε,i = t F¯ε,i (dt) 0 ∫ ∞ → e¯0,i = t F¯0,i (dt) < ∞ as ε → 0, (6.78) 0

where the limiting first moment e¯0,i = e0,ii , for i ∈ X, since the limiting probability p0,ii = 1. Also, by Lemma 2.2, for i ∈ X, ∫ ∞ eε,i,+ = tFε,i,+ (dt) 0 ∫ ∞ → e¯0,i,+ = t F¯0,i,+ (dt) = e¯0,i as ε → 0. (6.79) 0

122

6 Super-long and long time ergodic theorems for singularly perturbed ARP

Therefore, the condition Q1 holds for the regenerative processes ξε,i (t), for i ∈ X. The condition R2 implies that, for s ∈ U[q ·,i (·, A)], A ∈ Γ, i ∈ X, us

qε,i (·, A) −→ q0,i (·, A) as ε → 0.

(6.80)

According to the condition R2 , the function q0,i (t, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − F0,i,+ (t) = 1 − F0,i (t), t ∈ R+ , for i ∈ X. Also, according to the condition R2 , m(U¯ [q ·,i (·, A)]) = 0 for A ∈ Γ, i ∈ X. Finally, according to the condition R2 , the function q0,i (t, A), t ∈ R+ is continuous almost everywhere with respect to the Lebesgue measure on B+ , for A ∈ Γ, i ∈ X. Therefore, the conditions P¯ 2 , R2 , and U1 imply that the condition R1 is satisfied for the functions qε,i (t, A), t ∈ R+, A ∈ BZ (with the corresponding limiting function q0,i (·, A), sets of local uniform convergence U[q ·,i (·, A)], and the class Γ), for i ∈ X. Let s ∈ (0, ∞). We choose an arbitrary 0 ≤ sε → s as ε → 0. This relation obviously implies that tε = sε vε → ∞. The conditions U1 , Sβ , and the relations (6.69), (6.70) imply that, for i ∈ X, j  i, qε,i,+ tε = pε,i j sε vε −1 = sε pε,i j (p−1 ε,12 + pε,21 ) → sci,β as ε → 0.

(6.81)

Thus, all conditions of Theorem 2.3 are satisfied for the regenerative processes ξε,i (t), t ≥ 0 with the regenerative times ζε,i,n, n = 1, 2, . . . and the regenerative lifetimes με,i,+ , for i ∈ X. Therefore, the following relation holds, for any A ∈ Γ, s ∈ (0, ∞), and i ∈ X: Pε,i,+ (sε vε, A) = q˜ε,vε ,i (sε, A) → q˜0,i (s, A) = e−sci, β /e0, i π0,i (A) as ε → 0. The proof is complete.

(6.82) 

Let us also summarise the above remarks in the form of the following useful lemma. Lemma 6.6 Let the conditions O2 , P¯ 2 , Q2 , R2 , U1 , and Sβ (for some β ∈ [0, ∞]) be satisfied for the alternating regenerative processes (ξε (t), ηε (t)) with the regeneration times ζε,n . Then the condition R2 is satisfied for the functions q˜ε,vε ,i (t, A), t ∈ R+, A ∈ BZ, i ∈ X, with the limiting functions q˜0,i (t, A) = e−tci, β /e0, i π0,i (A), t ∈ R+, A ∈ BZ, i ∈ X, the class of set Γ ⊆ BZ , which appear in condition R2 , and sets U[q˜ ·,v·,i (·, A)] = (0, ∞), A ∈ Γ, i ∈ X, that is: (a) the function q˜0,i (t, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − F˜0,i (t) = e−tci, β /e0, i , t ∈ R+ ; (b) us q˜ε,vε ,i (·, A) −→ q˜0,i (·, A) as ε → 0, for s ∈ U[q˜ ·,v·,i (·, A)] = (0, ∞), A ∈ Γ, i ∈ X; (c) m(U¯ [q˜ ·,v·,i (·, A)]) = m({0}) = 0, for A ∈ Γ, i ∈ X; (d) the function q˜0,i (·, A) is continuous on the interval R+ , for A ∈ Γ, i ∈ X.

6.2 Super-long time ergodic theorems

123

6.2 Super-Long Time Ergodic Theorems and Embedded Regenerative Processes In this section, we describe the asymptotic behaviour for the probabilities Pε,i j (tε , A) for the so-called super-long times 0 ≤ tε → ∞ as ε → 0 satisfying the following relation: (6.83) tε /vε → ∞ as ε → 0. The corresponding limits for the stationary probabilities for the perturbed semiMarkov processes ηε,vε (t) take, for β ∈ [0, ∞], the following form: ρ1 (β) = where

e0,1 α1 (β) e0,2 α2 (β) , ρ2 (β) = , e(β) e(β)

(6.84)

−1 −1 , α2 (β) = c2,β , α1 (β) = c1,β

(6.85)

e(β) = e0,1 α1 (β) + e0,2 α2 (β).

(6.86)

ρ1 (β), ρ2 (β) ∈ (0, 1).

(6.87)

ρ1 (β) = 1, ρ2 (β) = 0.

(6.88)

ρ1 (β) = 0, ρ2 (β) = 1.

(6.89)

and If β ∈ (0, ∞), then, If β = 0, then, If β = ∞, then,

The corresponding limiting stationary probabilities for singularly perturbed alternating regenerative processes take the following form, for β ∈ [0, ∞]: (β)

π0, j (A) = ρ j (β)π0, j (A), for A ∈ BZ, j ∈ X.

(6.90)

It is useful to note that the above limiting probabilities coincide with the corresponding limiting probabilities for regularly perturbed alternating regenerative processes with parameter β = p0,12 /p0,21 given in the relations (4.50), (4.81), and (4.82). The following theorem takes place. Theorem 6.1 Let the conditions O2 , P¯ 2 , Q2 , R2 , U1 , and Sβ (for some β ∈ [0, ∞]) be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → ∞ as ε → 0, (β)

Pε,i j (tε, A) → π0, j (A) as ε → 0.

(6.91)

Proof We shall prove that all conditions of Theorem 2.1 are satisfied for the regenerative processes ξˇε,vε (t) = (ξε,vε (t), ηε,vε (t)) = (ξε (tvε ), ηε (tvε )), t ≥ 0, with the regeneration times ζˇε,vε ,n = vε−1 ζˇε,n, n = 0, 1, . . ..

124

6 Super-long and long time ergodic theorems for singularly perturbed ARP

We shall also prove that all conditions of Theorem 2.2 are satisfied for the versions of the regenerative processes ξˇε,vε (t) = (ξε,vε (t), ηε,vε (t)) = (ξε (tvε ), ηε (tvε )), t ≥ 0,   , n = 0, 1, . . . and the transition periods = vε−1 ζˇε,n with the regeneration times ζˇε,v ε ,n  [0, ζˇε,vε ,1 ). Regenerative lifetimes are not involved. We can use the Theorems 2.1–2.3, for the model with the stopping probabilities qε = 0, ε ∈ [0, 1]. Let us analyse the asymptotic behaviour of the probabilities Pˇε,1 (t, B). In this case, we do prefer to consider ξˇε,vε (t) = (ξε,vε (t), ηε,vε (t)), t ≥ 0 as a standard regenerative process with regeneration times 0 = ζˇε,vε ,0, ζˇε,vε 1, . . ., and the initial state ηε,vε (0) = 1. The renewal equation (6.5) for the probabilities Pˇε,vε ,1 (t, B) takes the following form, for B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ : ∫ t (6.92) Pˇε,vε ,1 (t, B) = qˇε,vε ,1 (t, B) + Pˇε,vε ,1 (t − s, B)Fˇε,vε ,1 (ds), t ≥ 0, 0

where, for t ≥ 0, Fˇε,vε ,1 (t) = P1 { ζˇε,vε ,1 ≤ t} = P1 { ζ˜ε,vε ,2 ≤ t}

(6.93)

and, for B ∈ BZ , t ≥ 0, qˇε,vε ,1 (t, B) = P1 { ξˇε,vε (t) ∈ B, ζˇε,vε ,1 > t} = P1 {(ξε (tvε ), ηε (tvε )) ∈ B, ζ˜ε,vε ,2 > t}.

(6.94)

Obviously, ζˇε,n ≥ ζε,n , for n = 0, 1, . . .. Thus, the condition O2 obviously implies that the condition O1 (a) is satisfied. The condition O1 (b) is also satisfied, since, the stopping probability qˇε,vε = 1 − Fˇε,vε ,1 (∞) ≡ 0. In this case, for t ≥ 0, Fˇε,vε ,1 (t) = P1 { ζ˜ε,vε ,2 ≤ t} = F˜ε,vε ,1 (t) ∗ F˜ε,vε ,2 (t),

(6.95)

eˇε,vε ,1 = E1 ζ˜ε,vε ,2 = e˜ε,vε ,1 + e˜ε,vε ,2 .

(6.96)

and The conditions P2 and U1 imply, by Lemmas 6.1 and 6.2, that the condition P1 (a) is satisfied for the regenerative processes ξˇε,vε (t), i.e., Fˇε,vε ,1 (·) → Fˇ0,1 (·) = P{

e0,1 e0,2 ζ1 + ζ2 ≤ ·} as ε → 0. c1,β c2,β

(6.97)

The distribution function Fˇ0,1 (·) is obviously non-arithmetic and, thus, the condition P1 (b) is also satisfied. The condition P1 (c) is satisfied, since the stopping probability qˇε,vε = 1 − Fˇε,vε ,1 (∞) ≡ 0. The conditions P¯ 2 , Q2 , and U1 imply, by Lemmas 6.3–6.4, that the condition Q1 is satisfied for the regenerative processes ξˇε,vε (t), i.e.,

6.2 Super-long time ergodic theorems

125

eˇε,vε ,1 = e˜ε,vε ,1 + e˜ε,vε ,2 → eˇ0,1 = e˜0,1 + e˜0,2 as ε → 0.

(6.98)

In this case, the function qˇε,vε ,1 (t, B) given by the relation (6.16) takes the following form: for t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ , ∫ t (6.99) qˇε,vε ,1 (t, B) = q˜ε,vε ,1 (t, A1 ) + F˜ε,vε ,1 (ds)q˜ε,vε ,2 (s, A2 ), 0

and the corresponding limiting function qˆ0,1 (t, B) given by the relation (6.21) takes the following form, for t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ : ∫ t (6.100) qˇ0,1 (t, B) = q˜0,1 (t, A1 ) + F˜0,1 (ds)q˜0,2 (s, A2 ), 0

where F˜0,1 (t) and q0, j (t, A j ), i = 1, 2 are the corresponding limiting quantities appearing in Lemmas 6.1 and 6.6. As is shown in Sect. 6.1.1.2, the condition R1 (a) is satisfied for the function qˇ0,1 (t, B), t ∈ R+, B = (A1 × {1}) ∪ (A2 × {2}) ∈ BZ . Let us consider sets B = A × {1}, A ∈ Γ. In this case, for t ∈ R+ , Pˇε,vε ,1 (t, A × {1}) = P1 {ξε,vε (t) ∈ A, ηε,vε (t) = 1} = Pε,vε ,11 (t, A), and the renewal equation (6.92) takes the following form: ∫ t Pε,vε ,11 (t, A) = qˇε,vε ,11 (t, A) + Pε,vε ,11 (t − s, A)Fˇε,vε ,1 (ds), t ≥ 0,

(6.101)

(6.102)

0

where, for t ∈ R+, A ∈ BZ , qˇε,vε ,11 (t, A) = qˇε,vε ,1 (t, A × {1}) = P1 { ξˇε,vε (t) ∈ A × {1}, ζˇε,vε ,1 > t} = P1 {ξε,vε (t) ∈ A, ηε,vε (t) = 1, ζ˜ε,vε ,2 > t}.

(6.103)

If ηε,vε (0) = 1, then ηε,vε (t) = 1, for t ∈ [0, ζ˜ε,vε ,1 ), and ηε,vε (t) = 2, for t ∈ [ζ˜ε,vε ,1, ζ˜ε,vε ,2 ). Therefore, for every t ∈ R+, A ∈ BZ , qˇε,vε ,11 (t, A) = P1 {ξε,vε (t) ∈ A, ηε,vε (t) = 1, ζ˜ε,vε ,2 > t} = P1 {ξε,vε (t) ∈ A, ζ˜ε,vε ,1 > t} = q˜ε,vε ,1 (t, A).

(6.104)

The relation (6.104) and Lemma 6.6 imply that the conditions R1 (b)–(d) are satisfied for the functions qˇε,vε ,11 (·, A) = q˜ε,vε ,1 (t, A), for A ∈ Γ, that is: (1) The functions us

qˇε,vε ,11 (·, A) = q˜ε,vε ,1 (·, A) −→ q˜0,1 (·, A) as ε → 0, for s ∈ U[q˜ ·,v·,1 (·, A)], A ∈ Γ, (2) m(U¯ [q˜ ·,v·,1 (·, A)]) = 0, A ∈ Γ, (3) the function q˜0,1 (·, A) is continuous almost everywhere with respect to Lebesgue measure on B+ , for A ∈ Γ.

126

6 Super-long and long time ergodic theorems for singularly perturbed ARP

As stated above, the stopping probabilities qˇε,vε ≡ 0, and, thus, qˇε,vε tε → 0 as ε → 0, for any 0 ≤ tε → ∞ as ε → 0. Thus, all conditions of Theorem 2.1 hold for the embedded regenerative processes ξˇε,vε (t) and, thus, the ergodic relation given in this theorem takes place for the probabilities Pε,vε ,11 (tε , A) = Pε,11 (tε vε, A), for any A ∈ Γ and 0 ≤ tε → ∞ as ε → 0, Pε,vε ,11 (tε , A) = Pε,11 (tε vε, A) (β)

→ π0,1 (A) = =

1 eˇ0,11

∫

∞

0

1 e0,1 /c1,β + e0,2 /c2,β

q˜0,1 (s, A)ds ∫ ∞ e−sc1, β /e0,1 π0,1 (A)ds as ε → 0.

(6.105)

0 (β)

The expressions for the probabilities π0,1 (A) given in the relations (6.90) and (6.105) coincide, for A ∈ Γ. Indeed, ∫ ∞ 1 e−sc1, β /e0,1 π0,1 (A)ds e0,1 /c1,β + e0,2 /c2,β 0 e0,1 /c1,β (β) = π0,1 (A) = π0,1 (A). (6.106) e0,1 /c1,β + e0,2 /c2,β Thus, the following ergodic relation holds, for A ∈ Γ and 0 ≤ tε → ∞ as ε → 0: (β)

Pε,11 (tε vε, A) → π0,1 (A) as ε → 0.

(6.107)

Let us now consider the compressed version of the regenerative process ξˇε,vε (t)  ). It is the regenerative process, with the transition period [0, ζˇε,v ε ,1 ξˇε,vε (t) = (ξε,vε (t), ηε,vε (t)), t ≥ 0 = (ξε (tvε ), ηε (tvε )), t ≥ 0,

(6.108)

with the regeneration times,   ζε,v = vε−1 ζˇε,n , n = 0, 1, . . . . ε ,n

(6.109)

  + t), ηε,vε (ζˇε,v + t)), t ≥ 0 is a standard regenThe shifted process (ξε,vε (ζˇε,v ε ,1 ε ,1 erative process.  ) = 1. Therefore, the probabilities Pε,vε ,11 (t, A) If ηε,vε (0) = 2, then ηε,vε (ζˇε,v ε ,1

play for this process the role of the probabilities Pε(1) (t, A) introduced in Sect. 2.1.1.2. The conditions P¯ 2 , Q2 , Sβ , U1 , and Lemma 6.1 imply that the condition P 1 is  ≤ ·}, with the satisfied for the distribution functions Fε (·) = F˜ε,vε ,2 (·) = P2 { ζˇε,v ε ,1  ˜ corresponding limiting distribution function F0 (·) = F0,2 (·) = P{e0,2 /c2,β ζ ≤ ·}.

6.2 Super-long time ergodic theorems

127

Thus, all conditions of Theorem 2.2 hold, and, thus, the ergodic relation (6.107) for the probabilities Pε,vε ,11 (tε , A) = Pε,11 (tε vε, A) also holds for the probabilities Pε,vε ,21 (tε , A) = Pε,21 (tε vε, A), i.e., for any A ∈ Γ and 0 ≤ tε → ∞ as ε → 0, (β)

Pε,vε ,21 (tε , A) = Pε,21 (tε vε, A) → π0,1 (A) as ε → 0.

(6.110)

Due to the symmetry of the conditions O2 , P¯ 2 , Q2 , R2 , Sβ , and U1 with respect to the indices i, j = 1, 2, the ergodic relations, similar to the above-mentioned ergodic relations for the probabilities Pε,vε ,11 (tε , A) = Pε,11 (tε vε, A) and Pε,vε ,21 (tε , A) = Pε,21 (tε vε, A), also take place for the probabilities Pε,vε ,22 (tε , A) = Pε,22 (tε vε, A) and Pε,vε ,12 (tε , A) = Pε,12 (tε vε, A). They take the following forms, for any i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0: (β)

Pε,vε ,i2 (tε , A) = Pε,i2 (tε vε, A) → π0,2 (A) as ε → 0.

(6.111)

The above analysis, in particular, the relations (6.107), (6.110), and (6.111) give a description of the asymptotic behaviour of the probabilities Pε,i j (tε, A) for superlong times 0 ≤ tε → ∞ as ε → 0 satisfying the asymptotic relation tε /vε → ∞ as ε → 0. To see this, one should just represent such tε in the form, tε = tε vε .  Obviously, tε = tε /vε → ∞ as ε → 0. There exists also an alternative approach for obtaining super-long ergodic theorems for singularly perturbed alternating regenerative processes. Let us assume that ηε,vε (0) = 1 and consider the alternating regenerative process ζ˜ε,ve,1 (t) = (ξε,vε (t), ηε,vε (t)) with the regeneration times ζ˜ε,vε ,n = vε−1 ζ˜ε,n . This process is a standard alternating regenerative process, since its semi-Markov component ηε,vε (t) = 1 for t ∈ [ζ˜ε,vε ,2n, ζ˜ε,vε ,2n+1 ), n = 0, 1, . . ., while ηε,vε (t) = 2 for t ∈ [ζ˜ε,vε ,2n+1, ζ˜ε,vε ,2n+2 ), n = 0, 1, . . .. It is easy to see that the embedded regenerative process of the second type ζˇε,ve,1 (t) = (ξε,vε (t), ηε,vε (t)) with the regeneration times ζˇε,vε ,n = vε−1 ζˇε,n coincides with the embedded regenerative process of the first type, ξˆ˜ε,vε (t) = (ξε,vε (t), ηε,vε (t)) with the regeneration times ζˆ˜ε,vε ,n = ζ˜ε,vε ,2n . Indeed, the regenerative process ζˇε,ve,1 (t) = ξˆ˜ε,vε (t) = (ξε,vε (t), ηε,vε (t)) has regeneration times ζˇε,vε ,n = ζˆ˜ε,vε ,n = ζ˜ε,vε ,2n . The conditions O2 , P¯ 2 , Q2 R2 , U1 , Sβ , and Lemmas 6.1–6.6 imply that: (1) The conditions O2 , P2 , Q2 , and R2 are satisfied for the standard alternating regenerative processes ζ˜ε,ve,1 (t), if β = 0, (2) the conditions O2 , P2 , Q2 , and R2 are satisfied for the standard alternating regenerative processes ζ˜ε,ve,1 (t), if β ∈ (0, ∞), and (3) the conditions O2 , P2, Q2 , and R2 are satisfied for the standard alternating regenerative processes ζ˜ε,ve,1 (t), if β = ∞. In all three cases, the corresponding limiting quantities are given in Lemmas 6.1– 6.6. . The above remarks make it possible to obtain the basic ergodic relation (6.107) by applying to the standard alternating regenerative processes ζ˜ε,ve,1 (t) = (ξε,vε (t),

128

6 Super-long and long time ergodic theorems for singularly perturbed ARP

ηε,vε (t)) with the regeneration times ζ˜ε,vε ,n Theorem 5.1, in case β = 0, Theorem 4.1, in case β ∈ (0, ∞), or Theorem 5.2, in case β = ∞. The rest of the proof repeats the corresponding part of the proof (after the relation (6.107)) of Theorem 6.1.

6.3 Long Time Ergodic Theorems for Singularly Perturbed Alternating Regenerative Processes In this subsection, we describe the asymptotic behaviour of the probabilities Pε,i j (tε , A) for the so-called long times 0 ≤ tε → ∞ as ε → 0, which satisfy the following asymptotic relation: (6.112) tε /vε → t ∈ (0, ∞) as ε → 0. Let β ∈ (0, ∞), and η(β) (t), t ≥ 0 be a continuous time homogeneous Markov chain with the phase space Y = {1, 2}, the transition probabilities of embedded Markov chain pi j = I(i  j), i, j = 1, 2, and the distribution functions of sojourn (β) (β) times in states 1 and 2, respectively, F1 (t) = 1 − e−tc1, β /e0,1 , t ≥ 0 and F2 (t) = 1 − e−tc2, β /e0,2 , t ≥ 0. We also assume that this Markov chain has continuous from the right trajectories. Consider the transition probabilities for the Markov chain η(β) (t), (β)

pi j (t) = Pi {η(β) (t) = j}, t ≥ 0, i, j ∈ Y.

(6.113)

(β)

Explicit expressions for the transition probabilities pi j (t) are well known, as solutions of the corresponding forward Kolmogorov system of differential equations (β) for these probabilities. Namely, the corresponding 2 × 2 matrix P(β) (t) = pi j (t) has the following form, for t ≥ 0:    ρ (β) + ρ2 (β)e−λ(β)t ρ2 (β) − ρ2 (β)e−λ(β)t  , (6.114) P(β) (t) =  1 ρ1 (β) − ρ1 (β)e−λ(β)t ρ2 (β) + ρ1 (β)e−λ(β)t  where λ1 (β) = c1,β /e0,1, λ2 (β) = c2,β /e0,2, λ(β) = λ1 (β) + λ2 (β), and ρ1 (β) =

λ2 (β) e0,1 /c1,β λ1 (β) e0,2 /c2,β = , ρ2 (β) = = , λ(β) e(β) λ(β) e(β)

(6.115)

(6.116)

where e(β) = e0,1 /c1,β + e0,2 /c2,β .

(6.117)

Note that the Markov chain η(β) (t) is ergodic and ρi (β), i = 1, 2 are its stationary probabilities.

6.3 Long time ergodic theorems

129 (β)

The corresponding limiting probabilities π0,i j (t, A) for alternating regenerative processes (ξε (t), ηε (t)) take, in this case, the following forms, for t ∈ (0, ∞) and β ∈ [0, ∞]:  π0,1 (A) for A ∈ BZ, j = 1, i ∈ X, (0) (0) (6.118) (t, A) = π (A) = π0,i j 0, j 0 for A ∈ BZ, j = 2, i ∈ X,  0 for A ∈ BZ, j = 1, i ∈ X, (∞) (∞) π0,i (6.119) (t, A) = π (A) = j 0, j π0,2 (A) for A ∈ BZ, j = 2, i ∈ X, and, for β ∈ (0, ∞), (β)

(β)

π0,i j (t, A) = pi j (t)π0, j (A) for A ∈ BZ, j = 1, 2, i ∈ X.

(6.120)

The following theorem takes place. Theorem 6.2 Let the conditions O2 , P¯ 2 , Q2 R2 , U1 be satisfied and, also, the condition Sβ is satisfied for some β ∈ [0, ∞]. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → t ∈ (0, ∞) as ε → 0, (β)

Pε,i j (tε, A) → π0,i j (t, A) as ε → 0.

(6.121)

Proof The renewal equation (6.102) for the compressed regenerative process, (ξε,vε (t), ηε,vε (t)), t ≥ 0 = (ξε (tvε ), ηε (tvε )), t ≥ 0, with the regeneration times, ζˇε,vε ,n = vε−1 ζˇε,n, n = 0, 1, . . ., takes, due to the relation (6.104), the following form: ∫ t Pε,vε ,11 (t, A) = q˜ε,vε ,11 (t, A) + Pε,vε ,11 (t − s, A)Fˇε,vε ,1 (ds), t ≥ 0. (6.122) 0

As well known, the solution of this equation has the form, ∫ t Pε,vε ,11 (t, A) = q˜ε,vε ,1 (t − s, A)Uˇ ε,vε ,11 (ds), t ≥ 0,

(6.123)

0

where Uˇ ε,vε ,1 (t) =

∞

n=0

(∗n) (t), t ≥ 0, Fˇε,v ε ,1

(6.124)

is the corresponding renewal function. The relation (6.38) given in Lemma 6.2 implies that, for n = 0, 1, . . ., (∗n) (∗n) (·) ⇒ Fˇ0,1 (·) as ε → 0. Fˇε,v ε ,1

(6.125)

(∗n) n The inequality Fˇε,v (t) ≤ Fˇε,v (t) obviously holds, for t ≥ 0 and n ≥ 1. These ε ,1 ε ,1 inequalities and the relation (6.51) imply that, for t ≥ 0 and n ≥ 1, (∗n) n n lim Fˇε,v (t) ≤ lim Fˇε,v (t) = Fˇ0,1 (t) < 1, ε ,1 ε ,1

ε→0

since, for t ≥ 0.

ε→0

(6.126)

130

6 Super-long and long time ergodic theorems for singularly perturbed ARP

Fˇ0,1 (t) = P{

e0,1 e0,2 ζ1 + ζ2 ≤ t} < 1. c1,β c2,β

(6.127)

Thus, the series on the right hand side in (6.124) converge asymptotically uniformly, as ε → 0. The above remarks imply that, for t > 0, Uˇ ε,vε ,1 (t) → Uˇ 0,1 (t) = Uˇ 0,1 (t) =

∞

(∗n) (t) as ε → 0. Fˇ0,1

(6.128)

n=0 (∗n) Since Fˇ0,1 (t) is a continuous distribution function, for n ≥ 1, the function Uˇ 0,1 (t) is also continuous, for t > 0. Thus, the convergence relation in (6.128) holds for all t > 0. The relation (6.74) given in Lemma 6.5 implies that, for every t > 0, us

q˜ε,vε ,1 (t − ·, A) −→ q˜0,1 (t − ·, A) as ε → 0 for s ∈ [0, t).

(6.129)

At the same time, due to continuity function Uˇ 0,1 (t), for t > 0, measure Uˇ 0,1 (ds) has no atom at any point s > 0. By the above remarks, relations (6.74), (6.128), Lemma B.2 imply that the following relation holds, for A ∈ Γ and t > 0: ∫ t Pε,vε ,11 (t, A) = q˜ε,vε ,1 (t − s, A)Uˇ ε,vε ,1 (ds) 0 ∫ t q˜0,1 (t − s, A)Uˇ 0,1 (ds) → P0,11 (t, A) = 0 ∫ t e−(t−s)c1, β /e0,1 Uˇ 0,1 (ds) as ε → 0. (6.130) = π0,1 (A) 0

Next, we make an important remark that the time compression transformation with the compression factor vε and all asymptotic relations presented above, in particular (6.125)–(6.126) and (6.128)–(6.130), can be, in obvious way, repeated using slightly modified compression factor v ε = aε vε , for any 0 < aε → 1 as ε → 0. In particular, the modified asymptotic relation (6.130) takes the following form, for A ∈ Γ and t > 0: Pε, vε ,11 (t, A) = Pε,11 (taε vε, A) → P0,11 (t, A) as ε → 0.

(6.131)

Thanks to an arbitrary choice of 0 < aε → 1 as ε → 0, the relation (6.131) for every t > 0 is equivalent to the following relation, which holds for any A ∈ Γ and 0 ≤ tε → t as ε → 0: Pε,11 (tε vε, A) = Pε,vε ,11 (tε, A) → P0,11 (t, A) as ε → 0.

(6.132)

6.3 Long time ergodic theorems

131

  The shifted process (ξε,vε (ζˇε,v + t), ηε,vε (ζˇε,v + t)), t ≥ 0 is a standard ε ,1 ε ,1 regenerative process.  ) = 1. That is why, the probabilities Pε,vε ,11 (t, A) If ηε,vε (0) = 2, then ηε,vε (ζˇε,v ε ,1

play for this process the role of the probabilities Pε(1) (t, A) introduced in Sect. 2.1.1.2. The distribution function for the duration of transition period,  F˜ε,vε ,2 (t) = P2 { ζˇε,v ≤ ·}. ε ,1

(6.133)

According to the relation (6.37) given in Lemma 6.1, the distribution functions, F˜ε,vε ,2 (·) ⇒ F˜0,2 (·) = P{

e0,2 ζ ≤ ·} as ε → 0. c2,β

(6.134)

If β ∈ (0, ∞], then F˜0,2 (t) = 1 − e−tc2, β /e0,2 , t ≥ 0 is an exponential distribution function. If β = 0, then F˜0,2 (t) = I(t ≥ 0), t ≥ 0. In both cases, F˜0,2 (t) is continuous function for t > 0. The renewal type transition relation (2.13) takes the following form: ∫ t Pε,vε ,21 (t, A) = Pε,vε ,11 (t − s, A)F˜ε,vε ,2 (ds), t ≥ 0. (6.135) 0

The relation (6.132) implies that, for every A ∈ Γ, t > 0, us

Pε,vε ,11 (t − ·, A) −→ P0,11 (t − ·, A) as ε → 0, for s ∈ [0, t).

(6.136)

At the same time, due to continuity of the distribution function F˜0,2 (t) for t > 0, the measure F˜0,2 (ds) has no atom at any point s > 0. Lemma B.2 implies, by the above remarks, the relation (6.37) given in Lemma 6.1, and the relation (6.136), that the following relation holds, for A ∈ Γ and t > 0: ∫ t Pε,vε ,21 (t, A) = Pε,vε ,11 (t − s, A)F˜ε,vε ,2 (ds) 0 ∫ t P0,11 (t − s, A)F˜0,2 (ds) = P0,21 (t, A). (6.137) → 0

Using arguments similar to those used for relations (6.130)–(6.132), it is possible to improve the relation (6.137) to a more advanced relation that holds, for A ∈ Γ and any 0 ≤ tε → t ∈ (0, ∞) as ε → 0, Pε,21 (tε vε, A) = Pε,vε ,21 (tε, A) → P0,21 (t, A) as ε → 0.

(6.138)

It remains to give more explicit expressions for the limiting probabilities P0,11 (t, A), t > 0 and P0,21 (t, A), t > 0.

132

6 Super-long and long time ergodic theorems for singularly perturbed ARP

First, let us consider the case, where β = 0. In this case, Fˇ0,1 (t) = P{e0,1 ζ1 ≤ t} = 1 − e−t/e0,1 , t ≥ 0, i = 1, 2 is an exponential 1 t, t ≥ 0. distribution function. Thus, the renewal function Uˇ 0,1 (t) = I(t ≥ 0) + e0,1 ˜ Also, the distribution function F0,2 (t) = I(t ≥ 0), t ≥ 0. Finally, c1,0 = 1, and the function q˜0,1 (s, A) = e−s/e0,1 π0,1 (A), s ∈ [0, ∞). Taking into account the above remarks, we get the following relations, for A ∈ BZ and t > 0: ∫ t e−(t−s)/e0,1 Uˇ 0,11 (ds) P0,11 (t, A) = π0,1 (A) 0

= π0,1 (A)(e−t/e0,1 +

∫

t

0

= π0,1 (A)(e

−t/e0,1

+e

e−(t−s)/e0,1 ds) e0,1

−t/e0,1

(et/e0,1 − 1)) = π0,1 (A)

(6.139)

and ∫ P0,21 (t, A) =

t

P0,11 (t − s, A)F˜0,2 (ds)

0

= P0,11 (t, A) = π0,1 (A).

(6.140)

Second, let us consider the case, where β = ∞. In this case, Fˇ0,1 (t) = P{e0,2 ζ2 ≤ t} = 1 − e−t/e0,2 , t ≥ 0 is an exponential 1 t, t ≥ 0. distribution function. Thus, the renewal function Uˇ 0,1 (t) = I(t ≥ 0) + e0,2 −t/e 0,2 ˜ Also, the distribution function F0,2 (t) = P{e0,2 ζ2 ≤ t} = 1 − e , t ≥ 0. Finally, c1,∞ = ∞ and function q˜0,1 (s, A) = 0, s ∈ [0, ∞). Therefore, for A ∈ BZ and t > 0, P0,11 (t, A) = P0,21 (t, A) = 0.

(6.141)

Third, let us consider the main case, where β ∈ (0, ∞). Let η(β) (t), t ≥ 0 be a continuous time homogeneous Markov chain introduced in the beginning of this section. (β) (β) (β) (β) Also, let ζn = inf(t > ζn−1, η(β) (t)  η(β) (ζn−1 )), n = 1, 2, . . . , ζ0 = 0 be successive moments of jumps for the Markov chain η(β) (t). The Markov chain η(β) (t) is also an alternating regenerative process with regen(β) eration times ζ2n , n = 0, 1, . . .. (β)

Let us assume that η(β) (0) = 1. The transition probabilities p11 (t), t ≥ 0 satisfy the following renewal equation:

6.3 Long time ergodic theorems (β)

133

∫

(β)

p11 (t) = q1 (t) +

t

0

(β)

(β)

p11 (t − s)F11 (ds), t ≥ 0,

(6.142)

where (β)

(β)

q1 (t) = P1 {η(β) (t) = 1, ζ2 =

(β) P1 {ζ1

> t}

> t} = e−tc1, β /e0,1 , t ≥ 0,

(6.143)

and (β)

(β)

F11 (t) = P1 {ζ2

≤ t}

(β)

(β)

= F1 (t) ∗ F2 (t), t ≥ 0,

(6.144)

where, for i = 1, 2, (β)

(β)

Fi (t) = Pi {ζ1

≤ t} = 1 − e−tci, β /e0, i , t ≥ 0.

(6.145)

Let us consider the corresponding renewal function generated by the distribution (β) function F11 (t), ∞

(β) (β)(∗n) U11 (t) = F11 (t), t ≥ 0. (6.146) n=0 (β)

The relation (6.52), given in Lemma 6.2, and relation (6.144) imply that F11 (·) = Fˇ0,v0,1 (·). Therefore, the corresponding renewal functions also coincide, i.e., for t ≥ 0, (β)

U11 (t) =

∞

(β)(∗n)

F11

n=0

(t) =

∞

(∗n) (t) = U0,1 (t). F˜0,1

(6.147)

n=0 (β)

The transition probabilities p11 (t) can be expressed, as the solution of the renewal equation (6.142), in the following form: ∫ t (β) (β) (β) p11 (t) = q1 (t)U11 (ds) 0 ∫ t e−(t−s)c1, β /e0,1 Uˇ 0,11 (ds). (6.148) = 0

Finally, the relations (6.130) and (6.148) imply that the following equality takes place, for A ∈ BZ and t > 0: (β)

P0,11 (t, A) = p11 (t)π0,1 (A).

(6.149)

Obviously, the following renewal type relation takes place for the transition probabilities of the Markov chain η(β) (t):

134

6 Super-long and long time ergodic theorems for singularly perturbed ARP (β)

p21 (t) =

∫ 0

t

(β)

(β)

p11 (t − s)F2 (ds), t ≥ 0.

(6.150)

(β) The distribution function F2 (t) = 1 − e−tc2, β /e0,2 = F˜0,2 (t), t ≥ 0. This equality and the relations (6.137), (6.149), and (6.150) imply that, for A ∈ BZ and t > 0, ∫ t P0,21 (t, A) = P0,11 (t − s, A)F˜0,2 (ds) 0 ∫ t (β) (β) p11 (t − s)π0,1 (A)F2 (ds) = 0 (β)

= p21 (t)π0,1 (A).

(6.151)

Due to the symmetry of the conditions O2 , P¯ 2 , Q2 , Sβ , and U1 with respect to the indices i, j = 1, 2, the ergodic relations, similar to the above-mentioned ergodic relations for the probabilities Pε,vε ,11 (tε, A) = Pε,11 (tε vε, A) and Pε,vε ,21 (tε, A) = Pε,21 (tε vε, A), also take place for the probabilities Pε,vε ,22 (tε, A) = Pε,22 (tε vε, A) and Pε,vε ,12 (tε, A) = Pε,12 (tε vε, A). The above analysis, in particular, relations (6.132), (6.138) supplemented by the explicit expression for the corresponding limiting probabilities gives a description of the asymptotic behaviour of the probabilities Pε,i j (tε, A) for long times 0 ≤ tε → ∞ as ε → 0 satisfying the asymptotic relation tε /vε → t ∈ (0, ∞) as ε → 0. To see this, one should just represent such tε in the form, tε = tε vε . Obviously, tε = tε /vε → t as ε → 0. 

Chapter 7

Short Time Ergodic Theorems for Singularly Perturbed Alternating Regenerative Processes

In this chapter, we present short time ergodic theorems for perturbed alternating regenerative processes based on two types of time compression factors. This chapter includes two sections. In Sect. 7.1, we consider singularly perturbed alternating regenerative processes and present short time ergodic asymptotics based on the first type time compression factor, in Theorem 7.1. In Sect. 7.2, we consider singularly perturbed alternating regenerative processes and present short time ergodic asymptotics based on the second type time compression factor, in Theorems 7.2–7.6.

7.1 Short Time Ergodic Theorems Based on the First Time Compression Factor In this section, we introduce two types of time compression factors for perturbed alternating regenerative processes and present short time individual ergodic theorems for singularly perturbed alternating regenerative processes based on the first type time compression factor.

7.1.1 Two Types of Time Compression Factors for Perturbed Alternating Regenerative Processes Let us introduce a function, wε = (pε,12 + pε,21 )−1, ε ∈ (0, 1].

(7.1)

This function has useful asymptotic properties different from the asymptotic properties of the function, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0_7

135

136

7 Short time ergodic theorems for singularly perturbed ARP −1 vε = p−1 ε,12 + pε,21, ε ∈ (0, 1].

(7.2)

We call the functions vε and wε as the first and second time compression factors, respectively. The symbols fε ∼ fε as ε → 0 and fε ≺ fε as ε → 0 are used for two functions 0 < fε, fε → ∞ as ε → 0 in the sense that, respectively, fε/ fε → 1 as ε → 0 and fε/ fε → 0 as ε → 0. The following lemma presents some useful relationships between the time compression factors vε and wε . Lemma 7.1 Let the condition U1 be satisfied. Then, 0 < wε < vε < ∞, ε ∈ [01], and: (i) If for some β ∈ (0, ∞) the condition Sβ holds, then vε ∼ p−1 ε,12 (1 + β) ∼ −1 −1 −1 −1 −1 −1 pε,21 (1 + β ) as ε → 0, while wε ∼ pε,12 (1 + β ) ∼ pε,21 (1 + β)−1 as ε → 0, and, thus, wε ∼

β v (1+β)2 ε

as ε → 0.

−1 (ii) If the condition S0 is satisfied, then vε ∼ p−1 ε,12 as ε → 0, and wε ∼ pε,21 ≺ vε as ε → 0. −1 (iii) If the condition S∞ is satisfied, then vε ∼ p−1 ε,21 as ε → 0, and wε ∼ pε,12 ≺ vε as ε → 0.

From the statement (i) of Lemma 7.1 it follows that, if the condition Sβ is satisfied for some β ∈ (0, ∞), relations tε /vε → t and tε /wε → t as ε → 0 generate, for each t ∈ [0, ∞], equivalent in a sense, asymptotic time zones. From the statements (ii) and (iii) of Lemma 7.1 it follows that, if the condition S0 or S∞ is satisfied, relations tε /vε → t and tε /wε → t as ε → 0 generate, for each t ∈ [0, ∞], essentially different asymptotic time zones. Taking these remarks into account, we assume that in the case where the condition S0 or S∞ is satisfied, the “short” times 0 ≤ tε → ∞ as ε → 0 satisfy, in addition to the asymptotic relation (7.4) given below, the following asymptotic relation: tε /wε → t ∈ [0, ∞] as ε → 0.

(7.3)

7.1.2 Ergodic Theorems Based on the First Time Compression Factor In this subsection, we describe the asymptotic behaviour of the probabilities Pε,i j (tε , A) for the so-called short times 0 ≤ tε → ∞ as ε → 0 satisfying the following asymptotic relation: (7.4) tε /vε → 0 as ε → 0. We also assume that, in addition to the condition U1 , the condition Sβ holds for some β ∈ (0, ∞).

7.1 Short time ergodic theorems based on the first time compression factor

137

The corresponding limiting probabilities for the alternating regenerative processes (ξε (t), ηε (t)) in this case are the same for any β ∈ (0, ∞) and take the following form:  π (A) for A ∈ BZ, j = i, i ∈ X, (7.5) π¯0,i j (A) = I( j = i)π0,i (A) = 0,i 0 for A ∈ BZ, j  i, , i ∈ X. The following theorem takes place. Theorem 7.1 Let the conditions O2 , P¯ 2 , Q2 , R2 , U1 be satisfied and, also, the condition Sβ be satisfied for some β ∈ (0, ∞). Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → 0 as ε → 0, Pε,i j (tε, A) → π¯0,i j (A) as ε → 0.

(7.6)

Proof Let us first analyse the asymptotic behaviour of the probabilities Pε,11 (tε, A) and, thus, assume that ηε (0) = 1. We return to the original alternating regenerative process (ξε (t), ηε (t)), t ≥ 0 with the regeneration times ζε,n, n = 0, 1, . . .. Recall the stopping time ζ˜ε,1 introduced in Sect. 3.3.2.3. In this case, this is the time of first hitting in the state 2 by the process ηε (t). Consider again the regenerative process ξε,1 (t), t ≥ 0 with the regeneration times ζε,1,n, n = 0, 1, . . ., and the random lifetime με,1,+ , which were introduced in Sect. 6.1.2.5. It is easy to see that for every t ≥ 0, F˜ε,1 (t) = P1 { ζ˜ε,1 ≤ t} = P{με,1,+ ≤ t}

(7.7)

and, for every A ∈ BZ, t ≥ 0, P1 {ξε (t) ∈ A, ηε (t) = 1, ζ˜ε,1 > t} = P{ξε,1 (t) ∈ A, με,1,+ > t}.

(7.8)

According to the relation (6.37) given in Lemma 6.1, if ηε (0) = 1, the random variables, e0,1 d ζ as ε → 0, (7.9) vε−1 ζ˜ε,1 −→ c1,β where ζ is a random variable exponentially distributed with parameter 1. Since, we assumed that tε /vε → 0 as ε → 0, it follows from the relations (7.7) and (7.9) that, P{με,1,+ > tε } = P1 { ζ˜ε,1 > tε }

= P1 {vε−1 ζ˜ε,1 > tε vε−1 } → 1 as ε → 0.

(7.10)

The relations (7.8) and (7.10) imply that P1 {ξε (tε ) ∈ A, ηε (tε ) = 1} − P1 {ξε (tε ) ∈ A, ηε (tε ) = 1, ζ˜ε,1 > tε }

≤ P1 { ζ˜ε,1 ≤ tε } → 0 as ε → 0

(7.11)

138

7 Short time ergodic theorems for singularly perturbed ARP

and similarly, P{ξε,1 (tε ) ∈ A} − P{ξε,1 (tε ) ∈ A, με,1,+ > tε }

≤ P{με,1,+ ≤ tε } → 0 as ε → 0.

(7.12)

Note that the conditions O2 , P¯ 2 , and R2 imply that the conditions O1 , P1 (with the stopping probabilities qε,1 ≡ 0), and R1 are satisfied for the regenerative processes ξε,1 (t). Therefore, the above relations and Theorem 2.1 (applied to the regenerative processes ξε,1 (t)), imply that, for every A ∈ Γ, lim P11 (tε, A) = lim P1 {ξε (tε ) ∈ A, ηε (tε ) = 1}

ε→0

ε→0

= lim P1 {ξε (tε ) ∈ A, ηε (tε ) = 1, ζ˜ε,1 > tε } ε→0

= lim P{ξε,1 (tε ) ∈ A, με,1,+ > tε } ε→0

= lim P{ξε,1 (tε ) ∈ A} = π0,1 (A). ε→0

(7.13)

Let us now analyse the asymptotic behaviour of the probabilities Pε,21 (t, A) and, thus, assume that ηε (0) = 2. In this case, the relation (6.37) given in Lemma 6.1 implies that the random variables, e0,2 d ζ as ε → 0, (7.14) vε−1 ζ˜ε,1 −→ c2,β where ζ is a random variable exponentially distributed with parameter 1. Since, we assumed that tε /vε → 0 as ε → 0, the relation (7.14) obviously implies that, P2 { ζ˜ε,1 > tε } = P2 {vε−1 ζ˜ε,1 > tε vε−1 } → 1 as ε → 0. (7.15) If ηε (0) = 2, then, for each t > 0, event {ηε (t) = 1} ⊆ { ζ˜ε,1 ≤ t}. Thus, for every A ∈ Γ, P21 (tε, A) = P2 {ξε (tε ) ∈ A, ηε (tε ) = 1}

≤ P2 { ζ˜ε,1 ≤ tε } → 0 as ε → 0.

(7.16)

Due to the symmetry of the conditions O2 , P¯ 2 , Q2 , R2 , and U1 with respect to the indices i, j = 1, 2, ergodic relations similar to the above-mentioned ergodic relations for the probabilities Pε,11 (tε, A) and Pε,21 (tε, A), also hold for the probabilities  Pε,22 (tε, A) and Pε,12 (tε, A).

7.2 Short time ergodic theorems based on the second time compression factor

139

7.2 Short Time Ergodic Theorems Based on the Second Time Compression Factor In this section, we present short time individual ergodic theorems for singularly perturbed alternating regenerative processes based on the second type time compression factor.

7.2.1 First Type Short Time Ergodic Theorems Based on the Second Time Compression Factor In this subsection, we consider the case, where β = 0 or β = ∞, and the parameter t = ∞, in relation (7.3). In this case, the relations (7.3) and (7.4) mean that, wε ≺ tε ≺ vε as ε → 0. The corresponding limiting probabilities take the following forms:  π (A) for A ∈ BZ, j = 1, π0,(0)j (A) = 0,1 0 for A ∈ BZ, j = 2, 

and π0,(∞) j (A)

=

0 for A ∈ BZ, j = 1, π0,2 (A) for A ∈ BZ, j = 2.

(7.17)

(7.18)

(7.19)

It is useful to note that the above limiting probabilities π0,(0)j (A) and π0,(∞) j (A) coincide with the corresponding limiting probabilities for semi-regularly perturbed alternating regenerative processes, respectively, given in the relation (4.81) (for the case where β = 0) and in the relation (4.82) (for the case where β = ∞). The following theorems take place. Theorem 7.2 Let the conditions O2 , P¯ 2 , Q2 , R2 , U1 , and S0 be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → 0 as ε → 0 and tε /wε → ∞ as ε → 0, Pε,i j (tε, A) → π0,(0)j (A) as ε → 0.

(7.20)

Theorem 7.3 Let the conditions O2 , P¯ 2 , Q2 , R2 , U1 , and S∞ be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → 0 as ε → 0 and tε /wε → ∞ as ε → 0, Pε,i j (tε, A) → π0,(∞) j (A) as ε → 0. Proof First, we prove Theorem 7.2.

(7.21)

140

7 Short time ergodic theorems for singularly perturbed ARP

It can be noted that the analysis of the asymptotic behaviour of the probabilities Pε,11 (tε , A) can be carried out absolutely analogously to those presented in the relations (7.7)–(7.13), in the proof of Theorem 7.1. The only difference is that the parameter β = 0, and, thus, the limiting random variable in the analogue of the asymptotic relation (7.9) has the form e0,1 ζ, where ζ is a random variable exponentially distributed with parameter 1. This analysis leads to the following asymptotic relation that holds, for every A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → 0 as ε → 0: Pε,11 (tε, A) → π0,1 (A) as ε → 0.

(7.22)

The asymptotics of the probabilities Pε,21 (tε, A) differs in those presented in Theorem 7.1. In fact, the asymptotic relation similar to (7.14) does not hold. The relation (6.37) given in Lemma 6.1, implies that, if β = 0 and ηε (0) = 2, the random variables, d (7.23) vε−1 ζ˜ε,1 −→ 0 as ε → 0. This asymptotic relation does not entail the fulfilment of a relation similar to (7.10). The correct normalisation function for the random variables ζ˜ε,1 in this case is wε ∼ p−1 ε,21 as ε → 0. According to this asymptotic equivalence relation and the relation (6.49) given in the proof of Lemma 6.1, if β = 0 and ηε (0) = 2, then, d

wε−1 ζ˜ε,1 −→ e0,2 ζ as ε → 0,

(7.24)

where ζ is a random variable exponentially distributed, with parameter 1. The probabilities Pε,11 (tε, A) and Pε,21 (tε, A) are connected by the following renewal type relation: ∫ tε Pε,21 (tε, A) = Pε,11 (tε − s, A)P2 { ζ˜ε,1 ∈ ds} 0

∫ =

0

∫ =

0

tε /wε ∞

Pε,11 (tε − swε, A)P2 {wε−1 ζ˜ε,1 ∈ ds}

Pε,11 (tε − swε, A)P2 {wε−1 ζ˜ε,1 ∈ ds},

(7.25)

where the function Pε,11 (tε − swε, A) is defined as 0, for tε − swε < 0. Let us choose an arbitrary sε → s ∈ [0, in f t y) as ε → 0. Obviously, (tε − sε wε )/wε = tε /wε − sε → ∞ and, thus, (tε − sε wε ) → ∞ as ε → 0. Also, (tε − sε wε )/vε = tε /vε − sε wε /vε → 0 as ε → 0. Therefore, according to the relation (7.22), the following asymptotic relation holds, for A ∈ Γ and s ∈ [0, ∞): Pε,11 (tε − sε wε, A) → π0,1 (A) as ε → 0.

(7.26)

7.2 Short time ergodic theorems based on the second time compression factor

141

i.e., for any A ∈ Γ and s ∈ [0, ∞), us

Pε,11 (tε − ·wε, A) −→ π0,1 (A) as ε → 0.

(7.27)

The relations (7.24) and (7.27) imply, by Lemma B.2, that the following relation holds, for A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → 0 as ε → 0 and tε /wε → ∞ as ε → 0: ∫ ∞ Pε,21 (tε, A) → π0,1 (A)P{e0,2 ζ ∈ ds} 0 ∫ ∞ −1 s −1 −e0,2 = π0,1 (A)e0,2 e ds = π0,1 (A) as ε → 0. (7.28) 0

According to Lemma 3.4, the phase space Z ∈ Γ. Also, π0,1 (Z) = 1. Thus, the relations (7.22) and (7.28) imply that the following relation holds, for A ∈ Γ and i ∈ X: Pε,i2 (tε, A) ≤ Pε,i2 (tε, Z) = 1 − Pε,i1 (tε, Z) → 1 − π0,1 (Z) = 0 as ε → 0.

(7.29)

The proof of Theorem 7.2 is complete. The proof of Theorem 7.3 is absolutely analogous to the proof of Theorem 7.2, due to the symmetry conditions O2 , P¯ 2 , Q2 , R2 , and U1 with respect to indices i, j ∈ X. The only formula (7.18) for the corresponding limiting probabilities should be replaced by the formula (7.19). 

7.2.2 Second Type Short Time Ergodic Theorems Based on the Second Time Compression Factor In this subsection, we consider the case, where the parameter t ∈ (0, ∞) in the relation (7.3). In this case, the relation (7.3) means that, tε ∼ twε as ε → 0, where t ∈ (0, ∞).

(7.30)

According to the statements (ii) and (iii) of Lemma 7.1, if the condition S0 or S∞ holds, then wε ≺ vε as ε → 0, and, thus, the relation (7.30) implies that the “short” time relation (7.4) holds, i.e., tε /vε → 0 as ε → 0. The corresponding limiting probabilities for alternating regenerative processes (ξε (t), ηε (t)) take the following forms, for t ∈ (0, ∞):

142

7 Short time ergodic theorems for singularly perturbed ARP

⎧ ⎪ ⎪ ⎪ ⎨ ⎪

π0,1 (A) 0 A) = (1 − e−t/e0,2 )π0,1 (A) ⎪ ⎪ ⎪ ⎪ e−t/e0,2 π0,2 (A) ⎩

A ∈ BZ, j A ∈ BZ, j A ∈ BZ, j A ∈ BZ, j

= 1, i = 2, i = 1, i = 2, i

= 1, = 1, = 2, = 2,

(7.31)

⎧ e−t/e0,1 π0,1 (A) for A ∈ BZ, j ⎪ ⎪ ⎪ ⎨ ⎪ (1 − e−t/e0,1 )π0,2 (A) for A ∈ BZ, j (∞) π0,i j (t, A) = ⎪ 0 for A ∈ BZ, j ⎪ ⎪ ⎪ (A) for A ∈ BZ, j π 0,2 ⎩

= 1, i = 2, i = 1, i = 2, i

= 1, = 1, = 2, = 2.

(7.32)

(0) π0,i j (t,

for for for for

and

The following theorems take place. Theorem 7.4 Let conditions O2 , P¯ 2 , Q2 , R2 , U1 , and S0 be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → t ∈ (0, ∞) as ε → 0, (7.33) Pε,i j (tε, A) → πi(0) j (t, A) as ε → 0. Theorem 7.5 Let conditions O2 , P¯ 2 , Q2 , R2 , U1 , and S∞ be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and 0 ≤ tε → ∞ as ε → 0 such that tε /wε → t ∈ (0, ∞) as ε → 0, (∞) Pε,i j (tε, A) → π0,i j (t, A) as ε → 0.

(7.34)

Proof First, we prove Theorem 7.4. It can be noted, as in the proof of Theorem 7.2, that the analysis of asymptotic behaviour for probabilities Pε,11 (tε, A) can be carried out in exactly the same way with those presented in the relations (7.7)–(7.13) in the proof of Theorem 7.1. The only difference is that the parameter β = 0, and, thus, the limiting random variable in the analogue of the asymptotic relation (7.9) has the form, e0,1 ζ, where ζ is a random variable exponentially distributed with parameter 1. This analysis shows that an asymptotic relation similar to (7.22) holds, i.e., for A ∈ Γ and any tε /vε → 0 as ε → 0, Pε,11 (tε, A) → π0,1 (A) as ε → 0.

(7.35)

As in the proof of Theorem 7.1, the renewal type relation (7.25) connecting the probabilities Pε,11 (tε, A) and Pε,21 (tε, A) takes place. Let us choose an arbitrary sε → s ∈ [0, ∞) as ε → 0. Obviously, (tε −sε wε )/wε = tε /wε − sε → t − s as ε → 0. Thus, for t > s, the following relations hold, (tε − sε wε ) = (tε /wε − sε )wε → ∞ as ε → 0 and (tε − sε wε )/vε = (tε /wε − sε )wε /vε → 0 as ε → 0. Also, for t < s, the function (tε − sε wε ) = (tε /wε − sε )wε → −∞ for ε → 0. Therefore, according to the relation (7.22) and the definition of Pε,11 (tε −swε, A) = 0, for tε − swε < 0 in the relation (7.25), the following asymptotic relation holds, for A ∈ Γ and s  t:

7.2 Short time ergodic theorems based on the second time compression factor

Pε,11 (tε − sε wε, A) → π0,1 (A)I(t > s) as ε → 0,

143

(7.36)

i.e., for 0 ≤ s  t, us

Pε,11 (tε − ·wε, A) −→ π0,1 (A)I(t > ·) as ε → 0.

(7.37)

Note that the locally uniform convergence of the functions Pε,11 (tε − ·wε, A) as ε → 0 is not guarantied for s = t. However, the distribution function of the limiting random variable in the relation (7.24) is exponential and, therefore, it does not have an atom at any point t > 0. Therefore, the relations (7.24) and (7.37) imply, by Lemma B.2, that the following relation takes place, for A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → t ∈ (0, ∞) as ε → 0: ∫ ∞ Pε,21 (tε, A) = Pε,11 (tε − swε, A)P2 {wε−1 ζ˜ε,1 ∈ ds}, ∫0 ∞ −1 −s/e0,2 → π0,1 (A)I(t > s)e0,2 e ds 0

= (1 − e−t/e0,2 )π0,1 (A) as ε → 0.

(7.38)

It remains to analyse the asymptotics of the probabilities Pε,12 (tε, A) and Pε,22 (tε, A). As indicated in Sect. 3.1.1.2, the phase space Z ∈ Γ. Also, π0,1 (Z) = 1. Thus, the relation (7.22) implies that the following relation holds, for A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → t ∈ (0, ∞) as ε → 0: Pε,12 (tε, A) ≤ Pε,12 (tε, Z) = 1 − Pε,11 (tε, Z) → 1 − π0,1 (Z) = 0 as ε → 0.

(7.39)

Let us introduce random variables, με,2,n = κε,2,n I(ηε,2,n = 2), n = 1, 2, . . . .

(7.40)

Consider now a sequence of stochastic triplets, ξ¯ε,2,n = ξε,2,n (t), t ≥ 0 , κε,2,n, με,2,n , n = 1, 2, . . . ,

(7.41)

a regenerative process, ξε,2 (t) = ξε,2,n (t − ζε,2,n−1 ), for t ∈ [ζε,2,n−1, ζε,2,n ), n = 1, 2, . . . ,

(7.42)

with regeneration times, ζε,2,n = κε,2,1 + · · · + κε,2,n, n = 1, 2, . . . , ζε,2,0 = 0, and a random lifetime,

(7.43)

144

7 Short time ergodic theorems for singularly perturbed ARP

με,2,+ = ζε,2,νε,2 ,

(7.44)

where, νε,2 = min(n ≥ 1 : με,2,n < κε,2,n ) = min(n ≥ 1 : ηε,2,n = 1).

(7.45)

Let us also denote, for A ∈ Γ, t ≥ 0, Pε,2,+ (t, A) = P2 {ξε,2 (t) ∈ A, με,2,+ > t}.

(7.46)

In this case, the distribution functions, F¯ε,2 (t) = P{κε,2,1 ≤ t}, t ≥ 0,

(7.47)

and Fε,2,+ (t) = P{κε,2,1 ≤ t, με,2,1 ≥ κε,2,1 } = P{κε,2,1 ≤ t, ηε,2,1 = 2}, t ≥ 0.

(7.48)

The corresponding renewal equation for the probabilities Pε,2,+ (t, A), t ≥ 0 has the following form: ∫ t Pε,2,+ (t, A) = qε,2 (t, A) + Pε,2,+ (t − s, A)Fε,2,+ (ds), t ≥ 0, (7.49) 0

where qε,2 (t, A) = P{ξε,2 (t) ∈ A, ζε,2,1 ∧ με,2,+ > t} = P{ξε,1 (t) ∈ A, ζε,2,1 > t}, t ≥ 0.

(7.50)

It is used here that P{με,2,+ ≥ ζε,2,1 } = 1, according to the relations (7.44) and (7.45). The corresponding stopping probability, qε,2,+ = P{με,2,1 < κε,2,1 } = P{ηε,2,1 = 1} = pε,21,

(7.51)

e¯ε,2 = E κε,2,1 = eε,21 + eε,22,

(7.52)

and the expectations, and eε,2,+ = E κε,2,1 I(με,2,1 ≥ κε,2,1 ) = E κε,2,1 I(ηε,2,1 = 2) = eε,22 . The following relation obviously takes place, for A ∈ BX, t ≥ 0:

(7.53)

7.2 Short time ergodic theorems based on the second time compression factor

Pε,2,+ (t, A) = P{ξε,2 (t) ∈ A, με,2,+ > t} = P2 {ξε (t) ∈ A, ζ˜ε,1 > t}.

145

(7.54)

Lemma 7.2 Let the conditions O2 , P¯ 2 , Q2 , R2 , U1 , and S0 be satisfied. Then, the conditions O1 –R1 are satisfied for the regenerative processes ξε,2 (t), t ≥ 0, with the regenerative times ζε,2,n, n = 1, 2, . . ., and the regenerative lifetimes με,2,+ , and the following relation takes place for every A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0 such that pε,21 tε ∼ tε /wε → t ∈ (0, ∞) as ε → 0: P2 {ξε (tε ) ∈ A, ζ˜ε,1 > tε } = Pε,2,+ (tε, A)

→ e−t/e0,2 π0,2 (A) as ε → 0.

(7.55)

The proof of this lemma is similar to the proof of Lemma 6.5. The probabilities Pε,22 (tε, A) and Pε,12 (tε, A) are connected by the following renewal type equation: Pε,22 (tε, A) = P2 {ξε (tε ) ∈ A, ζ˜ε,1 > tε } ∫ tε Pε,12 (tε − s, A)P2 { ζ˜ε,1 ∈ ds}. +

(7.56)

0

The integral on the right hand side of the above relation can be represented as follows: ∫ tε Pε,12 (tε − s, A)P2 { ζ˜ε,1 ∈ ds} 0 ∫ ∞ = Pε,12 (tε − swε, A)P2 {wε−1 ζ˜ε,1 ∈ ds}, (7.57) 0

where the function Pε,12 (tε − swε, A) is defined as 0 for tε − swε < 0. Recall that we assume that tε /wε → t ∈ (0, ∞) as ε → 0. Let us choose an arbitrary sε → s ∈ [0, ∞) as ε → 0. In this case, as shown above, (tε − sε wε ) → ∞ as ε → 0, for t > s, while (tε − sε wε ) → −∞ as ε → 0, for t < s. Using the relation (7.39) and the definition of Pε,12 (tε −swε, A) = 0, for tε −swε < 0, in the relation (7.57), we get the following asymptotic relation that holds, for A ∈ Γ, s  t, and any sε → s ∈ [0, ∞) as ε → 0: Pε,12 (tε − sε wε, A) → 0 as ε → 0,

(7.58)

i.e., for any 0 ≤ s  t, us

Pε,12 (tε − ·wε, A) −→ 0 as ε → 0.

(7.59)

The relations (7.24) and (7.59) imply, by Lemma B.2, that the following relation takes place, for A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → t ∈ (0, ∞) as ε → 0:

146

7 Short time ergodic theorems for singularly perturbed ARP

∫ 0

∞

Pε,12 (tε − swε, A)P2 {wε−1 ζ˜ε,1 ∈ ds} ∫ ∞ −1 −s/e0,2 → 0 · e0,2 e ds = 0 as ε → 0.

(7.60)

0

The relation (7.55) given in Lemma 7.2, and the relations (7.56), (7.57), and (7.60) imply that the following relation holds for A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → t ∈ (0, ∞) as ε → 0: Pε,22 (tε, A) → e−t/e0,2 π0,2 (A) as ε → 0.

(7.61)

The proof of Theorem 7.4 is complete. The proof of Theorem 7.5 is absolutely similar to the proof of Theorem 7.4, due to the symmetry of the conditions O2 , P¯ 2 , Q2 , R2 , and U1 with respect to indices i, j ∈ X. The only formula (7.31) for the corresponding limiting probabilities should be replaced by the formula (7.32). 

7.2.3 Third Type Short Time Ergodic Theorems Based on the Second Time Compression Factor In this subsection, we consider the case, where the parameter t = 0 in the relation (7.3). In this case, the relation (7.3) means, for times tε → ∞ as ε → 0, that, tε ≺ wε as ε → 0.

(7.62)

The corresponding limiting probabilities are the same for both cases, where condition S0 or S∞ is satisfied, and take the following form:  π (A) for A ∈ BZ, j = i, i ∈ X, π¯0,i j (A) = 0,i (7.63) 0 for A ∈ BZ, j  i, i ∈ X. The following theorem takes place. Theorem 7.6 Let the conditions O2 , P¯ 2 , Q2 , R2 , U1 , and S0 or S∞ be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → 0 as ε → 0, (7.64) Pε,i j (tε, A) → π¯0,i j (A) as ε → 0. Proof First, assume that the condition S0 holds. The relation tε /wε → 0 as ε → 0 implies the relation tε /vε → 0 as ε → 0. This makes it possible to repeat the part of proof of Theorem 7.1 given in relations (7.7)–(7.13) and to obtain the asymptotic relation, P11 (tε, A) → π0,1 (A) as ε → 0.

(7.65)

7.2 Short time ergodic theorems based on the second time compression factor

147

In the case, where the condition S0 is satisfied and ηε (0) = 2, the relation (7.24) holds. Since, we assumed that tε /wε → 0 as ε → 0, the relation (7.24) implies that P2 { ζ˜ε,1 > tε } = P2 {wε−1 ζ˜ε,1 > tε wε−1 } → 1 as ε → 0.

(7.66)

If ηε (0) = 2, then, for each t > 0, event {ηε (t) = 1} ⊆ { ζ˜ε,1 ≤ t}. Thus, for every A ∈ Γ, P21 (tε, A) = P2 {ξε (tε ) ∈ A, ηε (tε ) = 1}

≤ P2 { ζ˜ε,1 ≤ tε } → 0 as ε → 0.

(7.67)

As indicated in Lemma 3.4, the phase space Z ∈ Γ. Also, π0,1 (Z) = 1. Thus, relation (7.65) implies that the following relation holds, for A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → 0 as ε → 0: Pε,12 (tε, A) ≤ Pε,12 (tε, Z) = 1 − Pε,11 (tε, Z) → 1 − π0,1 (Z) = 0 as ε → 0.

(7.68)

Finally, let us analyse the asymptotic behaviour of the probabilities Pε,22 (tε, A) and, thus, assume that ηε (0) = 2. Consider again the original alternating regenerative processes (ξε (t), ηε (t)), t ≥ 0 with the regeneration times ζε,n, n = 0, 1, . . . and the stopping time ζ˜ε,1 , which is the time of first hitting in state 1 by the process ηε (t). It is easy to see that, for every t ≥ 0, F˜ε,2 (t) = P2 { ζ˜ε,1 ≤ t} = P{με,2,+ ≤ t}

(7.69)

and, for every A ∈ BZ, t ≥ 0, P2 {ξε (t) ∈ A, ηε (t) = 2, ζ˜ε,1 > t} = P{ξε,2 (t) ∈ A, με,2,+ > t}.

(7.70)

Since, tε /wε → 0 as ε → 0, the relations (7.66) and (7.69) imply that, P{με,2,+ > tε } = P2 { ζ˜ε,1 > tε }

= P2 {wε−1 ζ˜ε,1 > tε wε−1 } → 1 as ε → 0.

(7.71)

The relations (7.8) and (7.71) imply that P2 {ξε (tε ) ∈ A, ηε (tε ) = 2}

− P2 {ξε (tε ) ∈ A, ηε (tε ) = 2, ζ˜ε,1 > tε } ≤ P2 { ζ˜ε,1 ≤ tε } → 0 as ε → 0 and, analogously,

(7.72)

148

7 Short time ergodic theorems for singularly perturbed ARP

P{ξε,2 (tε ) ∈ A} − P{ξε,2 (tε ) ∈ A, με,2,+ > tε }

≤ P{με,2,+ ≤ tε } → 0 as ε → 0.

(7.73)

These relations and Theorem 2.1 (applied to the regenerative processes ξε,2 (t)) imply that, for every A ∈ Γ, lim P22 (tε, A) = lim P2 {ξε (tε ) ∈ A, ηε (tε ) = 2}

ε→0

ε→0

= lim P2 {ξε (tε ) ∈ A, ηε (tε ) = 2, ζ˜ε,1 > tε } ε→0

= lim P{ξε,2 (tε ) ∈ A, με,2,+ > tε } ε→0

= lim P{ξε,2 (tε ) ∈ A} = π0,2 (A). ε→0

(7.74)

The proof of Theorem 7.5, in which the condition S∞ is replaced by the condition  S0 , is absolutely similar.

Chapter 8

Ergodic Theorems for Singularly Perturbed Alternating Regenerative Processes Compressed in Time

In this chapter we present individual ergodic theorems for singularly perturbed alternating regenerative processes compressed in time. This chapter includes two sections. In Sect. 8.1, we consider the model of singularly perturbed alternating regenerative processes with degenerating regeneration times. The corresponding ergodic asymptotics is presented in Theorems 8.1 and 8.2. In Sect. 8.2, we consider the model of compressed in time singularly perturbed alternating regenerative processes. We present more general perturbation conditions in Lemma 8.1, which let us generalise ergodic Theorems 6.1, 6.2, 7.1–7.5, and 8.1, 8.2 for singularly perturbed alternating regenerative processes on the model of singularly perturbed alternating regenerative processes compressed in time.

8.1 Singularly Perturbed Alternating Regenerative Processes with Degenerating Regeneration Times In this section, we present ergodic theorems for singularly perturbed alternating regenerative processes with degenerating regeneration times.

8.1.1 Super-Long and Long Time Ergodic Theorems In this section, we present ergodic theorems for singularly perturbed alternating regenerative processes for the case, where, either the distribution functions F0,1j (·), j ∈ Y0,1 or F0,2j (·), j ∈ Y0,2 degenerate, i.e., coincides with the distribution function F0 (u) = I(u ≥ 0), which has a unit jump in point 0. We assume that the condition U1 is satisfied. In this case, the conditions P2 and P2 can be simplified as this is described for the condition P2 in Sect. 5.1, and reduce them to the following simpler forms: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0_8

149

150

8 Ergodic theorems for singularly ARP compressed in time

P¯ 2 : Q ε,ii (·) ⇒ Q0,ii (·) as ε → 0, for i ∈ X, where Q0,11 (·) = F0,1 (·) is a weakly non-arithmetic distribution function, while Q0,22 (·) = F0,2 (·) = I(· ≥ 0) and P¯ 2: Q ε,ii (·) ⇒ Q0,ii (·) as ε → 0, for i ∈ X, where Q0,11 (·) = F0,1 (·) = I(· ≥ 0), while Q0,22 (·) = F0,2 (·) is a weakly non-arithmetic distribution function. The following two theorems take place. Theorem 8.1 Let the conditions O2 , P¯ 2 , Q2 , R2 , U1 , and Sβ (for some β ∈ [0, ∞)) be satisfied. Then, for A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0, such that tε /vε → t ∈ (0, ∞] as ε → 0, Pε,i j (tε, A) → π0,(0)j (A) as ε → 0.

(8.1)

Theorem 8.2 Let the conditions O2 , P¯ 2, Q2 , R2 , U1 , and Sβ (for some β ∈ (0, ∞]) be satisfied. Then, for A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0, such that tε /vε → t ∈ (0, ∞] as ε → 0, Pε,i j (tε, A) → π0,(∞) j (A) as ε → 0.

(8.2)

Proof Let us assume that the conditions U1 and Sβ (for some β ∈ [0, ∞)) are satisfied, and prove Theorem 8.1. Two cases should be considered. In the first case, limε→0 tε /vε = t = ∞. In this case, the ergodic relation (8.2) presents a modified variant of the super-long time ergodic relation given in Theorem 6.1. The first part of the proof repeats the corresponding part of the proof for Theorem 6.1, up to obtaining the ergodic relation (6.107). We apply Theorems 2.1–2.3 to the regenerative processes ξˇε,vε (t) = (ξε,vε (t), ηε,vε (t)) = (ξε (tvε ), ηε (tvε )), t ≥ 0 with the regeneration times ζˇε,vε ,n = vε−1 ζˇε,n, n = 0, 1, . . .. The probabilities Pε,vε ,11 (t, A) = Pε,11 (tvε, A) satisfy the renewal equation (6.102). We can repeat the corresponding part of the proof of Theorem 6.1 and obtain the following convergence relation (for the free term qˇε,1 (sε vε, A) in the renewal equation (6.102)) similar to the relation (6.74) given in Lemma 6.5, for A ∈ Γ and any 0 ≤ sε → s ∈ (0, ∞): qˇε,vε ,1 (sε, A) → qˇ0,1 (s, A) = e−sc1, β /e0,1 π0,1 (A) as ε → 0. Thus, for any s ∈ (0, ∞),

(8.3)

8.1 Singularly perturbed ARP with degenerating regeneration times

151

us

qˇε,vε ,1 (·, A) −→ qˇ0,1 (·, A) = e−· c1, β /e0,1 π0,1 (A) as ε → 0.

(8.4)

Also, we can repeat the corresponding part of the proof of Theorem 6.1 and obtain the following asymptotic relation similar to (6.105), for A ∈ Γ and any tε → ∞ as ε → 0: Pε,11 (tε vε, A) = Pε,vε ,11 (tε , A) → P0,11 (t, A) ∫ ∞ 1 = e−sc1, β /e0,1 π0,1 (A)ds eˇ0,1 0 e0,1 /c1,β = π0,1 (A) as ε → 0. eˇ0,1

(8.5)

The only difference between Theorems 6.1 and 8.1 lies in the form of the corresponding limiting distribution function Fˇ0,1 (·) and its first moment eˇ0,1 = ∫∞ ˇ s F0,1 (ds). 0 e e In Theorem 6.1, the distribution function Fˇ0,1 (·) = P{ c1,0,1β ζ1 + c2,0,2β ζ2 ≤ ·}, where ζ1 and ζ2 are two independent random variables exponentially distributed e with parameter 1. In Theorem 8.1, the distribution function Fˇ0,1 (·) = P{ c1,0,1β ζ1 ≤ ·}. Indeed, the conditions O2 , P¯ 2 , Q2 , U1 , and Sβ (assumed to be satisfied for some β ∈ [0, ∞]) imply, by Lemma 6.2, that Fˇε,vε ,1 (·) ⇒ Fˇ0,1 (·) as ε → 0. According to the condition P¯ 2 , both first moments e0,1, e0,2 > 0 and any value e e β ∈ [0, ∞] is admissible. The distribution function Fˇ0,1 (·) = P{ c1,0,1β ζ1 + c2,0,2β ζ2 ≤ ·} is not concentrated at zero. Obviously, it is a non-arithmetic distribution function. It is easy to see that the proof of Lemma 6.2 also admits the case, where e0,1 > 0, e0,2 = 0. This allows the condition P¯ 2 to be replaced by the condition P¯ 2 , in e Lemma 6.2. If β ∈ [0, ∞), the distribution function Fˇ0,1 (·) = P{ c1,0,1β ζ1 ≤ ·}. It is not concentrated at zero. It is an exponential distribution function that is non-arithmetic. If β = ∞, the distribution function Fˇ0,1 (·) = I(· ≥ 0). This case should be excluded. e e Respectively, in Theorem 6.1, the first moment eˇ0,1 = c1,0,1β + c2,0,2β , while in e

Theorem 8.1, the first moment eˇ0,1 = c1,0,1β . Also, in Theorem 6.1, the limiting stationary probability given by the relation (8.5) takes the following form, for β ∈ [0, ∞] and A ∈ Γ: P0,11 (t, A) =

e0,1 /c1,β π0,1 (A) e0,1 /c1,β + e0,2 /c2,β (β)

= ρ1 (β)π0,1 (A) = π0,1 (A).

(8.6)

While, in Theorem 8.1, the limiting stationary probability, given by the relation (8.5) takes the following form, for β ∈ [0, ∞) and A ∈ Γ:

152

8 Ergodic theorems for singularly ARP compressed in time

P0,11 (t, A) =

e0,1 /c1,β (0) π (A) = π0,1 (A). e0,1 /c1,β 0,1

(8.7)

Therefore, if the conditions of Theorem 8.1 are satisfied, then the following ergodic relation, which is an analogue to the ergodic relation (6.107), holds, for A ∈ Γ, and any 0 ≤ tε → ∞ as ε → 0: (0) (A) as ε → 0. Pε,11 (tε vε, A) = Pε,vε ,11 (tε , A) → π0,1

(8.8)

The process ξˇε,vε (t) = (ξε,vε (t), ηε,vε (t)) = (ξε (tvε ), ηε (tvε )), t ≥ 0 is also a re , n = 0, 1, . . . and the transition generative process with the regeneration times ζˇε,v ε ,n period [0, ζ˜ε,vε ,1 ). If ηε,vε (0) = 2, the distribution function for the duration of the transition period is Fε (·) = F˜ε,vε ,2 (·) = P2 { ζ˜ε,vε ,1 ≤ ·}. The conditions O2 , P¯ 2 or P¯ 2 , Q2 , Sβ (assumed to be satisfied for some β ∈ [0, ∞)), and U1 imply that the condition P 1 is satisfied, and the corresponding limiting e distribution F0 (·) = F˜0,2 (·) = P{ c2,0,2β ζ2 ≤ ·}. There is a difference between the cases where the condition P¯ 2 or P¯ 2 is satisfied. In the first case, F˜0,2 (·) is an exponential distribution function, and in the second case, F˜0,2 (·) = I(· ≥ 0). This case also is admissible in the condition P 1 . Thus, Theorem 2.2 can be applied to the above alternating regenerative processes. This gives, in the case when the conditions of Theorem 8.1 are satisfied, the following ergodic relation, for A ∈ Γ, and any 0 ≤ tε → ∞ as ε → 0: (0) (A) as ε → 0. Pε,21 (tε vε, A) = Pε,vε ,21 (tε , A) → π0,1

(8.9)

The relations (8.8) and (8.9) imply that, for A ∈ Γ, i = 1, 2, and any 0 ≤ tε → ∞ as ε → 0, Pε,i2 (tε vε, A) ≤ Pε,i2 (tε vε, Z)

= 1 − Pε,i1 (tε vε, Z)

(0) (Z) = 0 as ε → 0. → 1 − π0,1

(8.10)

The above analysis, in particular, the relations (8.8), (8.9), and (8.10) gives a description of the asymptotic behaviour of the probabilities Pε,i j (tε, A) for superlong times 0 ≤ tε → ∞ as ε → 0 satisfying the asymptotic relation tε /vε → ∞ as ε → 0. To verify this, it suffices to represent such tε in the form, tε = tε vε . Obviously, tε = tε /vε → ∞ as ε → 0. The proof of Theorem 8.1, for the case when limε→0 tε /vε = t = ∞ is complete. In the second case, limε→0 tε /vε = t ∈ (0, ∞). In this case, the ergodic relation (8.2) presents a modified variant of the long time ergodic relation given in Theorem 6.2. The first part of the proofs repeats the corresponding part of the proof for Theorem 6.2, up to getting of ergodic relation (6.132). This yields the following asymptotic relation, analogous to (6.132), for A ∈ Γ and any tε → t ∈ (0, ∞) as ε → 0:

8.1 Singularly perturbed ARP with degenerating regeneration times

Pε,11 (tε vε, A) = Pε,vε ,11 (tε, A) → P0,11 (t, A) ∫ t e−(t−s)c1, β /e0,1 Uˇ 0,1 (ds) as ε → 0, = π0,1 (A)

153

(8.11)

0

 ˇ (∗n) where Uˇ 0,1 (s) = ∞ n=0 F0,1 (s), s ≥ 0 is the renewal function for the corresponding limiting distribution function Fˇ0,1 (·). The difference between Theorems 6.2 and 8.1 is in the form of the corresponding limiting distribution Fˇ0,1 (·) generating the limiting renewal function Uˇ 0,1 (s). 1 ζ1 +e0,2 1+β1 −1 ζ2 ≤ ·}, In Theorem 6.2, the distribution function Fˇ0,1 (·) = P{e0,1 1+β where ζ1 and ζ2 are two independent random variables, exponentially distributed with parameter 1. In this case, any value β ∈ [0, ∞] is admissible. In Theorem 8.1, 1 ζ1 ≤ ·}. In this case, any value β ∈ [0, ∞) is the distribution Fˇ0,1 (·) = P{e0,1 1+β admissible. c In the latter case, the renewal function Uˇ 0,1 (s) = I(s ≥ 0) + e1,0,1β s, s ≥ 0 and, thus, for any β ∈ [0, ∞) and A ∈ Γ, ∫ t P0,11 (t, A) = π0,1 (A) e−(t−s)c1, β /e0,1 Uˇ 0,1 (ds) 0 ∫ t −tc1, β /e0,1

+ e−(t−s)c1, β /e0,1 (c1,β /e0,1 )ds = π0,1 (A) e 0

= π0,1 (A) e−tc1, β /e0,1 + e−tc1, β /e0,1 (etc1, β /e0,1 − 1) = π0,1 (A). (8.12) Therefore, if conditions of Theorem 8.1, the following ergodic relation holds, for A ∈ Γ, and any 0 ≤ tε → t as ε → 0: (0) (A) as ε → 0. Pε,11 (tε vε, A) = Pε,vε ,11 (tε, A) → π0,1

(8.13)

Also, we can repeat the corresponding part of the proof for Theorem 6.2 and get the following asymptotic relation, analogous to (6.138), for A ∈ Γ and any tε → t ∈ (0, ∞) as ε → 0: Pε,21 (tε vε, A) = Pε,vε ,21 (tε, A) → P0,21 (t, A) ∫ t P0,11 (t − s, A)F˜0,2 (ds) as ε → 0, =

(8.14)

0 e where the limiting distribution function F˜0,2 (·) = P{ c2,0,2β ζ2 ≤ ·}. The only difference between Theorems 6.2 and 8.1 is in the form of the distribution function F˜0,2 (·). In Theorem 6.2, the expectation e0,2 > 0, according to the condition P¯ 2 . Thus, ˜ F0,21 (·) is an exponential distribution function with the parameter c2,β /e0,2 , if β ∈ (0, ∞], or F˜0,21 (·) = F0 (·) = I(· ≥ 0), if β = 0.

154

8 Ergodic theorems for singularly ARP compressed in time

In Theorem 8.1, the expectation e0,2 = 0, according to the condition P¯ 2 . Thus, F˜0,2 (·) = F0 (·) = I(· ≥ 0), for any β ∈ [0, ∞). In the latter case, the relation (8.14) implies that, for every β ∈ [0, ∞), A ∈ Γ, and t > 0, ∫ t P0,11 (t − s, A)F0 (ds) P0,21 (t, A) = 0 (0 (A). = P0,11 (t, A) = π0,1

(8.15)

Therefore, if the conditions of Theorem 8.1 are satisfied, the following ergodic relation holds, for A ∈ Γ, and any 0 ≤ tε → t as ε → 0: (0) (A) as ε → 0. Pε,21 (tε vε, A) = Pε,vε ,21 (tε, A) → π0,1

(8.16)

The rest of the proof is similar to the corresponding part of the proof of Theorem 6.2. From the relations (8.13) and (8.16) it follows that, for A ∈ Γ, and any 0 ≤ tε → t ∈ (0, ∞) as ε → 0, Pε,i2 (tε vε, A) ≤ Pε,i2 (tε vε, Z) = 1 − Pε,i1 (tε vε, Z) (0) (Z) = 0 as ε → 0. → 1 − π0,1

(8.17)

The above analysis, in particular, the relations (8.13), (8.16), and (8.17) gives the description of the asymptotic behaviour of the probabilities Pε,i j (tε, A) for super-long times 0 ≤ tε → ∞ as ε → 0 satisfying the asymptotic relation tε /vε → t ∈ (0, ∞) as ε → 0. To see this, one should just represent such tε in the form, tε = tε vε . Obviously, tε = tε /vε → t as ε → 0. The proof of Theorem 8.1 for the case where limε→0 tε /vε = t ∈ (0, ∞) is complete. The proof of Theorem 8.2 is similar.  Remark 8.1 It is useful to note that in Theorem 8.1, i.e., if the conditions U1 and P¯ 2 are satisfied, the condition R2 can be weakened. In fact, the assumption of asymptotic locally uniform convergence of the functions qε,i (t, A) expressed in the asymptotic relation given in this condition may be required only for i = 1. The similar asymptotic relation for i = 2 follows from the inequalities qε,2 (t, A) ≤ 1 − Fε,2 (t), t ≥ 0 and the weak convergence relation Fε,2 (·) ⇒ I(· ≥ 0) as ε → 0. Similarly, in Theorem 8.2, i.e., in the case, where the conditions U1 and P¯ 2 are satisfied, the condition R2 can be weakened, and the assumption of asymptotic locally uniform convergence of the functions qε,i (t, A) expressed in the asymptotic relation given in this condition may be required only for i = 2. Remark 8.2 The case, where the conditions U1 and S∞ are satisfied, is excluded in Theorem 8.1. This is so, since in this case, it follows from the condition P¯ 2 that

8.1 Singularly perturbed ARP with degenerating regeneration times

155

the distribution functions of the regeneration times Fˇε,vε ,1 (·) ⇒ I(· ≥ 0) as ε → 0. Thus, Theorem 2.2 is not applicable. Remark 8.3 Similarly, the case, where the conditions U1 and S0 are satisfied, is excluded in Theorem 8.2. This is so, since in this case, it follows from the condition P¯ 2 that the distribution functions of regeneration times Fˇε,vε ,2 (·) ⇒ I(· ≥ 0) as ε → 0. Thus, Theorem 2.2 is not applicable.

8.1.2 Short Time Ergodic Theorems The following two theorems take place. Theorem 8.3 Let the conditions O2 , P¯ 2 , Q2 , R2 , U1 , and S0 be satisfied. Then, for A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0, such that tε /vε → 0 as ε → 0 and tε /wε → t ∈ (0, ∞] as ε → 0, Pε,i j (tε, A) → π0,(0)j (A) as ε → 0.

(8.18)

Theorem 8.4 Let the conditions O2 , P¯ 2, Q2 , R2 , U1 , and S∞ be satisfied. Then, for A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0, such that tε /vε → 0 as ε → 0, and tε /wε → t ∈ (0, ∞] as ε → 0, Pε,i j (tε, A) → π0,(∞) j (A) as ε → 0.

(8.19)

Proof First, we prove Theorem 8.3 for the case, where limε→0 tε /wε = ∞. It can be noted that the analysis of the asymptotic behaviour of the probabilities Pε,11 (tε , A) can be carried out in exactly the same way as it was done in the proofs of Theorems 7.1 and 7.2. The condition P¯ 2 can be used instead of the condition P¯ 2 , since only the characteristics of the stochastic triplets ξ¯ε,1,n = ξε,1,n (t), t ≥ 0 , κε,1,n, με,1,n , n = 1, 2, . . . were involved. Since the parameter β = 0, the limiting random variable in the analogue of the asymptotic relation (7.9) has the form, e0,1 ζ, where ζ is a random variable exponentially distributed with parameter 1. The above analysis gives the following asymptotic relation (similar to (7.22)), which holds, for A ∈ Γ and any tε /vε → 0 as ε → 0: Pε,11 (tε, A) → π0,1 (A) as ε → 0.

(8.20)

The asymptotic behaviour of the probabilities Pε,21 (tε, A) differs from those presented in Theorem 7.2. In fact, an asymptotic relation similar to (7.24) does not hold. Instead of this relation, from the conditions P¯ 2, Q2 , S0 , and Lemma 6.1 it follows that in the case when ηε (0) = 2, d wε−1 ζ˜ε,1 −→ ζ˜0,1 ≡ 0 as ε → 0,

(8.21)

156

8 Ergodic theorems for singularly ARP compressed in time

d In the relation (7.24), the random variables wε−1 ζ˜ε,1 −→ ζ˜0,1 as ε → 0, where the limiting random variable ζ˜0,1 has an exponential distribution function with parameter −1 . While, in the relation (8.21), the limiting random variable ζ˜ has a distribution e0,2 0,1 function concentrated at zero, i.e., P{ ζ˜0,1 ≤ t} = I(t ≥ 0), t ≥ 0. However, the rest of the proof is still similar to the corresponding part of the proof of Theorem 7.2. An asymptotic analysis repeating those given in the proof of Theorem 7.2 gives the following relation (similar to the relation (7.28)), which holds, for A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0, such that tε /vε → 0 as ε → 0 and tε /wε → ∞ as ε → 0: ∫ ∞ Pε,21 (tε, A) → π0,1 (A)P{ ζ˜0 ∈ ds} = π0,1 (A) as ε → 0. (8.22)

0

As indicated in Lemma 3.4, the phase space Z ∈ Γ. Also, π0,1 (Z) = 1. Thus, from the relations (8.20) and (8.22) it follows that the following relation holds, for A ∈ Γ and i ∈ X: Pε,i2 (tε, A) ≤ Pε,i2 (tε, Z) = 1 − Pε,i1 (tε, Z) → 1 − π0,1 (Z) = 0 as ε → 0.

(8.23)

The proof of Theorem 8.3 for the case, where limε→0 tε /wε = ∞, is complete. Second, let us prove Theorem 8.3 for the case, where limε→0 tε /wε = t ∈ (0, ∞). It can be noted that the analysis of the asymptotic behaviour of the probabilities Pε,11 (tε , A) can be carried out in exactly the same way as it was done in the proofs of Theorems 7.1 and 7.4. The condition P¯ 2 can be used instead of the condition P¯ 2 , since only the characteristic of the stochastic triplets ξ¯ε,1,n = ξε,1,n (t), t ≥ 0 , κε,1,n, με,1,n , n = 1, 2, . . . were involved. Since the parameter β = 0, the limiting random variable in the analogue of the asymptotic relation (7.9) has the form, e0,1 ζ, where ζ is a random variable exponentially distributed with parameter 1. The above analysis gives the following asymptotic relation (similar to (7.35)), which holds, for A ∈ Γ and any tε /vε → 0 as ε → 0: Pε,11 (tε, A) → π0,1 (A) as ε → 0.

(8.24)

Moreover, the asymptotic analysis repeating those given in the proof of Theorem 7.4 gives the following relation (similar to (7.37)), which holds, for A ∈ Γ, i, j ∈ X, 0 ≤ s  t, and any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → 0 as ε → 0, and tε /wε → t ∈ (0, ∞] as ε → 0: us

Pε,11 (tε − ·wε, A) −→ π0,1 (A)I(t > ·) as ε → 0.

(8.25)

The relations (8.21) and (8.25) imply, by Lemma B.2, that the following relation takes place, for A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → t ∈ (0, ∞) as ε → 0:

8.2 Compression in time for singularly perturbed ARP

∫ Pε,21 (tε, A) =

∞

0

∫

→

157

Pε,11 (tε − swε, A)P2 {wε−1 ζ˜ε,1 ∈ ds},

∞

π0,1 (A)I(t > s)P{ ζ˜0 ∈ ds}

0

= π0,1 (A) as ε → 0.

(8.26)

The final part of the proof repeats those given above for the relation (8.23). The proof of Theorem 8.3 is complete. The proof of Theorem 8.4 is similar.  Remark 8.4 It should be noted that analogues of Theorems 7.1 and 7.6 do not take place for the cases, where the condition P¯ 2 is replaced by the condition P¯ 2 or P¯ 2. In fact, the relation (7.9) or (7.24) implies the relation (7.15) or (7.66), respectively. While the relation (8.21) is not enough to get a relation similar to (7.15) or (7.66).

8.2 Compression in Time for Singularly Perturbed Alternating Regenerative Processes In this section, we describe the time compression procedure for singularly perturbed alternating regenerative processes, similar to the time compression procedure for regularly perturbed alternating regenerative processes described in Sect. 5.2. Let us assume that the condition U1 is satisfied, and the following perturbation condition (weaker than the condition Pˆ 2 formulated in Sect. 5.2) holds for the perturbed alternating regenerative processes (ξε (t), ηε (t)) with the regenerative times ζε,n : Pˆ¯ 2 : (a) Q ε,ii (· uε,i ) ⇒ Q0,ii (·) as ε → 0, for i, j ∈ X, where Q0,ii (·) = F0,i (·) is a weakly non-arithmetic distribution function, for i ∈ X, (b) uε,i ∈ (0, ∞), for ε ∈ (0, 1], i ∈ X and uε,i → u0,i ∈ (0, ∞] as ε → 0, for i ∈ X. The conditions U1 and Pˆ¯ 2 imply that the following relation holds, for t ≥ 0, j  i, i ∈ X: Q ε,i j (tuε,i ) ≤ pε,i j → 0 as ε → 0

(8.27)

and, thus, for i ∈ X, Fε,i (· uε,i ) = Q ε,ii (· uε,i ) + Q ε,i j (· uε,i ) ⇒ Q0,ii (·) = F0,i (·) as ε → 0.

(8.28)

ˆ 2 (formulated in Sect. 5.2) is satisfied. We also assume that the condition Q ˆ 2. In this case, the limiting quantities e0,i j = 0, j  i, i ∈ X in the condition Q Let, for ε ∈ (0, 1] and A ∈ BZ, i ∈ X,

158

8 Ergodic theorems for singularly ARP compressed in time

qε,i (tuε,i, A) = P{ξε,i (tuε,i ) ∈ A, ζε,i,1 > tuε,i ), t ≥ 0, i ∈ X,

(8.29)

where the regeneration process ξε,i (t) and its regeneration times ζε,i,n are defined by the relations (3.49) and (3.50). The function qε,i (·uε,i, A), t ∈ R+, A ∈ BZ belongs to the class P[BZ ], and it is consistent with the distribution function Fε,i (·uε,i ), for i ∈ X and ε ∈ (0, 1], i.e., qε,i (tuε,i, Z) = 1 − Fε,i (tuε,i ), for t ∈ R+ .

(8.30)

ˆ 2 (also formulated in Sect. 5.2) We also assume that the perturbation condition R is satisfied. As in Sect. 5.2, we use the time compression factor, uε = uε,1 + uε,2, ε ∈ (0, 1].

(8.31)

Condition Pˆ¯ 2 (b) implies that, uε → u0 = u0,1 + u0,2 ∈ (0, ∞] as ε → 0.

(8.32)

We also assume that the time compression factors uε,i, i ∈ X are asymptotically comparable in the sense that the condition Xγ is satisfied, that is, uε,2 /uε,1 → γ ∈ [0, ∞] as ε → 0.

(8.33)

Consider again the stochastic triplets ξ¯ε,uε ,i,n = ξε,uε ,n (t), t ≥ 0 , κε,uε ,i,n , ηε,uε ,i,n , i = 1, 2, n = 1, 2, . . . constructed in accordance with the time compression relation (3.96) and the compressed in time alternating regenerative process (ξε,uε (t), ηε,uε (t)) = (ξε (tuε ), ηε (tuε )), t ≥ 0 with the regeneration times ζε,uε ,n = u−1 ε ζε,n, n = 0, 1, . . ., defined by relations (3.97)–(3.99). The transition characteristics of the compressed in time alternating regenerative process (ξε,uε (t), ηε,uε (t)) and the original alternating regenerative process (ξε (t), ηε (t)) are connected by the following relations, for ε ∈ (0, 1]: Q ε,uε ,i j (t) = Q ε,i j (tuε ), t ≥ 0, i, j ∈ X, pε,uε ,i j = pε,i j = Q ε,i j (∞), i, j ∈ X, ∫ ∞ −1 eε,uε ,i j = u−1 e = u uQ ε,i j (du), i, j ∈ X, ε ε,i j ε 0

qε,uε ,i (t, A) = qε,i (tuε, A), t ≥ 0, A ∈ BZ, i ∈ X.

(8.34)

The conditions U1 , Pˆ¯ 2 and Xγ imply that the following relations take place, for i, j ∈ X: (8.35) pε,uε ,i j = pε,i j → p0,i j = I(i = j) as ε → 0, and Q ε,uε ,i j (·) = Q ε,i j (· uε,i where

uε (γ) ) ⇒ Q0,i j (·) as ε → 0, uε,i

(8.36)

8.2 Compression in time for singularly perturbed ARP

⎧ Q0,11 (· (1 + γ)) ⎪ ⎪ ⎪ ⎪ ⎪ I(· ≥ 0) ⎨ ⎪ (γ) Q0,i j (·) = I(· ≥ 0) ⎪ ⎪ ⎪ Q0,22 (· (1 + γ −1 )) ⎪ ⎪ ⎪ 0 ⎩

159

for γ ∈ [0, ∞), i = 1, for γ = ∞, i = 1, for γ = 0, i = 2, for i = 2, for j  i, i = 1, 2.

(8.37)

Therefore, the conditions U1 , Pˆ¯ 2 , Xγ and the relations (8.35) and (8.37) entail the fulfilment of the condition P¯ 2 , P¯ 2 , or P¯ 2 with the corresponding limiting distributions (γ) Q0,i j (·), i, j ∈ X (for the alternating regenerative processes (ξε,uε (t), ηε,uε (t))) if, respectively, γ = 0, γ ∈ (0, ∞), or γ = ∞. ˆ 2 , and Xγ imply that the following relation holds, for i, j ∈ X The conditions U1 , Q and γ ∈ [0, ∞]: eε,uε ,i j = where (γ)

e0,i j

eε,i j eε,i j uε,i (γ) = → e0,i j as ε → 0, uε uε,i uε

⎧ e0,11 (1 + γ)−1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨0 ⎪ = 0 ⎪ ⎪ e0,22 (1 + γ −1 )−1 ⎪ ⎪ ⎪ ⎪ 0 ⎩

for γ ∈ [0, ∞), i = 1, for γ = ∞, i = 1, for γ = 0, i = 2, for i = 2, for j  i, i = 1, 2.

(8.38)

(8.39)

ˆ 2 , Xγ and the relation (8.38) imply that condition Therefore, the conditions Q (γ) Q2 is satisfied, with the corresponding limiting expectations e0,i j , i, j ∈ X (for the alternating regenerative processes (ξε,uε (t), ηε,uε (t))), for γ ∈ [0, ∞]. Let us define sets, U[γ, q ·,u·,i (·, A)]

⎧ {s ≥ 0 : s(1 + γ) ∈ U[q ·,i (· u · ,i, A)]} ⎪ ⎪ ⎪ ⎨ [0, ∞) ⎪ = [0, ∞) ⎪ ⎪ ⎪ ⎪ {s ≥ 0 : s(1 + γ −1 ) ∈ U[q ·,i (· u · ,i, A)]} ⎩

for γ for γ for γ for γ

∈ [0, ∞), i = 1, = ∞, i = 1, = 0, i = 2, ∈ (0, ∞], i = 2.

(8.40)

ˆ 2 , m(U¯ [q · ,i (· u ·,i, A)]) = 0, for i ∈ X, A ∈ Γ and, According to the condition R therefore, m(U¯ [γ, q · ,u· ,i (· , A)]) = 0, for i ∈ X, A ∈ Γ and γ ∈ [0, ∞]. ˆ 2 and Xγ imply that the following relation takes place, for The conditions R s ∈ U[γ, q ·,u·, i ,i (· , A)], i ∈ X, A ∈ Γ and γ ∈ [0, ∞]: qε,uε ,i (·, A) = qε,i (· uε, A) us uε (γ) , A) −→ q0,i (·, A) as ε → 0, = qε,i (· uε,i uε,i where

(8.41)

160

8 Ergodic theorems for singularly ARP compressed in time

⎧ q0,1 (·(1 + γ), A) ⎪ ⎪ ⎪ ⎨ 0(·) ≡ 0 ⎪ (γ) q0,i (·, A) = 0(·) ≡ 0 ⎪ ⎪ ⎪ ⎪ q0,2 (·(1 + γ −1 ), A) ⎩

for γ for γ for γ for γ

∈ [0, ∞), i = 1, A ∈ BZ, = ∞, i = 1, A ∈ BZ, = 0, i = 2, A ∈ BZ, ∈ (0, ∞], i = 2, A ∈ BZ .

(8.42)

ˆ 2 , Xγ and the relation (5.50) imply that the condition Therefore, the conditions R (γ) R2 is satisfied, with the corresponding limiting functions q0,i (t, A), t ≥ 0, i ∈ X, A ∈ BZ and the convergence sets U¯ [γ, q · ,u· ,i (· , A)]), i ∈ X, A ∈ BZ (for the alternating regenerative processes (ξε,uε (t), ηε,uε (t))), for γ ∈ [0, ∞]. It is also useful to note that, according to the relations (8.39) and (8.42), the corresponding limiting stationary probabilities, ∫ ∞ 1 (γ) (γ) q0,i (s, A)m(ds) π0,i (A) = (γ) e0,i 0 ∫ ∞ 1 q0,i (s, A)m(ds) = π0,i (A), (8.43) = e0,i 0 do not depend on the parameter γ, in the cases, γ ∈ [0, ∞), i = 1, A ∈ BZ and γ ∈ (0, ∞], i = 2, A ∈ BZ . The following lemma summarises the above remarks. ˆ 2, R ˆ 2 , U1 , Xγ (for some γ ∈ [0, ∞]) be Lemma 8.1 Let the conditions O2 , Pˆ¯ 2 , Q satisfied for the alternating regenerative processes (ξε (t), ηε (t)). Then the following conditions are satisfied for the alternating regenerative processes (ξε,uε (t), ηε,uε (t)) compressed in time by the factor uε given by the relation (8.31): (i) If γ ∈ (0, ∞), the conditions O2 , P¯ 2 , Q2 , and R2 are satisfied. (ii) If γ = 0, the conditions O2 , P¯ 2 , Q2 , and R2 are satisfied. (iii) If γ = ∞, the conditions O2 , P¯ , Q2 , and R2 are satisfied. 2

(iv) The asymptotic relations (8.35), (8.36), (8.38), and (8.41) play the roles of asymptotic relations appearing in the above conditions. The function uε is given by relation (8.31). The corresponding limiting quantities and sets appearing in these conditions and related relations are given by the relations (8.34)–(8.43). Remark 8.5 Lemma 8.1 allows to apply to the alternating regenerative processes ˆ 2, (ξε,uε (t), ηε,uε (t)) Theorems 6.1, 6.2, 7.1, 8.1, and 8.2, if the conditions O2 , Pˆ¯ 2 , Q ˆ R2 , U1 , Xγ , and Sβ are satisfied, and tε /vε → t ∈ [0, ∞] as ε → 0. In particular, Theorem 6.1 can be applied, if γ ∈ (0, ∞), β ∈ [0, ∞], and t = ∞, Theorem 6.2, if γ ∈ (0, ∞), β ∈ [0, ∞], and t ∈ (0, ∞), Theorem 7.1, if γ ∈ (0, ∞), β ∈ (0, ∞), and t = 0, Theorem 8.1, if γ = 0, β ∈ [0, ∞), and t ∈ (0, ∞], Theorem 8.2, if γ = ∞, β ∈ (0, ∞], and t ∈ (0, ∞]. The cases, where the vector parameter (β, γ) take value (∞, 0) or (0, ∞), are excluded. Indeed, according to Remarks 8.2 and 8.3, Theorems 8.1 and 8.2 do not work in these cases, for the reasons indicated in the Remarks 8.2 and 8.3.

8.2 Compression in time for singularly perturbed ARP

161

Remark 8.6 Also, Lemma 8.1 allows to apply to the alternating regenerative processes (ξε,uε (t), ηε,uε (t)) Theorems 7.2–7.6 and 8.3, 8.4, if the conditions O2 , ˆ 2, R ˆ 2 , U1 , Xγ , and S0 or S∞ are satisfied, and tε /vε → 0 as ε → 0 and Pˆ¯ 2 , Q tε /wε → t ∈ [0, ∞] as ε → 0 (wε = o(vε ) as ε → 0, if the condition S0 or S∞ is satisfied, i.e., β = 0 or β = ∞). In particular, Theorem 7.2 can be applied, if γ ∈ (0, ∞), β = 0, and t = ∞, Theorem 7.3, if γ ∈ (0, ∞) and β = ∞, and t = ∞, Theorem 7.4, if γ ∈ (0, ∞), β = 0, and t ∈ (0, ∞), Theorem 7.5, if γ ∈ (0, ∞), β = ∞, and t ∈ (0, ∞), Theorem 7.6, if γ ∈ (0, ∞), β = 0 or β = ∞, and t = 0, Theorem 8.3, if γ = 0, β = 0, and t ∈ (0, ∞], Theorem 8.4, if γ = ∞, β = ∞, and t ∈ (0, ∞]. It is useful to note that the parameter t determining different asymptotic time zones in the ergodic Theorems 6.1, 6.2, 7.1–7.6, and 8.1–8.4 is defined as t = limε→0 tε /vε , in the super-long and long time ergodic Theorems 6.1, 6.2, 8.1, 8.2 and the short time ergodic Theorem 7.1, or as t = limε→0 tε /wε , in the short time ergodic Theorems 7.2–7.6, 8.3, and 8.4. Also worth mentioning is the case where it is assumed that there exists a limiting alternating regenerative process (ξ0 (t), η0 t), that is, it is assumed that there exist some stochastic triplets ξ¯0,i,n = ξ0,n (t), t ≥ 0 , κ0,i,n , η0,i,n , i = 1, 2, n = 1, 2, . . ., having the properties (F)–(J) and the alternating regenerative process (ξ0 (t), η0 t) is constructed using the above stochastic triplets and the following relations, similar to relations (3.1)–(3.3): ξ0 (t) = ξε,η0, n−1,n (t − ζ0,n−1 ) and η0 (t) = η0,n−1, for t ∈ [ζ0,n−1, ζ0,n ), n = 1, 2, . . . .

(8.44)

ζ0,n = κ0,1 + · · · + κ0,n, n = 1, 2, . . . , ζ0,0 = 0.

(8.45)

where, In this case, the transition characteristics of the limiting alternating regenerative process (ξ0 (t), η0 (t)) and the compressed in time alternating regenerative process ˆ 2, R ˆ 2 , U1 , X ˆ γ and the relations (ξε,uε (t), ηε,uε (t)) are defined in the conditions Pˆ¯ 2 , Q (8.35)–(5.54), and, thus, are connected by the following relations: (γ)

Q0,i j (t) = Q0,i j (t), t ≥ 0, i, j ∈ X, (γ)

p0,i j = Q0,i j (∞), i, j ∈ X, (γ)

e0,i j = e0,i j , i, j ∈ X, (γ)

q0,i (t, A) = q0,i (t, A), t ≥ 0, A ∈ BZ, i ∈ X.

(8.46)

Chapter 9

Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes

In this chapter, we present individual ergodic theorems for perturbed alternating regenerative processes modulated by two states super-singularly perturbed semiMarkov processes (shortly referred as super-singularly perturbed alternating regenerative processes). This is the case where one of regime switching probabilities for perturbed modulating semi-Markov process identically equal to zero, while another one is positive but converges to zero, as ε → 0. This chapter includes three sections. In Sect. 9.1, we consider super-singularly perturbed alternating regenerative processes and find the form of aggregation for regeneration times appropriate for the asymptotic ergodic analysis of such processes. The super-long time ergodic asymptotics is presented in Theorems 9.1 and 9.2, the long-time ergodic asymptotics, in Theorems 9.3 and 9.4, and the short time ergodic asymptotics, in Theorems 9.5. Also, the ergodic asymptotics for the extremal case of absolutely singularly perturbed alternating regenerative processes, where both switching probabilities for perturbed modulating semi-Markov process identically equal to zero, is described in Theorem 9.6. In Sect. 9.2, we consider the model of super-singularly perturbed alternating regenerative processes with degenerating regeneration times. The corresponding ergodic asymptotics for transition probabilities is presented in Theorems 9.7 and 9.8. Also, the model of super-singularly perturbed alternating regenerative processes compressed in time is considered. We present more general perturbation conditions in Lemma 9.1, which let us generalise ergodic Theorems 9.1–9.4 and 9.6–9.8 on the model of super-singularly perturbed alternating regenerative processes compressed in time. In Sect. 9.3, we present some generalisations of ergodic theorems for perturbed alternating regenerative processes on models of alternating regenerative processes with arbitrary initial distributions of modulating semi-Markov component, alternating regenerative processes with transition periods, and models with weaker nonarithmetic assumptions for limiting distributions of transition times for modulating semi-Markov component. In this section, we also present the complete classification of ergodic theorems for perturbed alternating regenerative processes based on 26 such theorems given in Chaps. 4–9. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0_9

163

164

9 Ergodic theorems for super-singularly perturbed ARP

9.1 Super-Long, Long, and Short Time Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes In this section, we present super-long, long, and short time ergodic theorems for super-singularly perturbed alternating regenerative processes. As for singularly perturbed alternating regenerative processes, these theorems take different forms for different asymptotic time zones.

9.1.1 Super-Singularly Perturbed Alternating Regenerative Processes These are alternating regenerative processes with a super-singular perturbation model, where, in addition to O2 –R2 , the following condition is satisfied: V1 : (a) pε,12 = 0, for ε ∈ (0, 1], and 0 < pε,21 → p0,21 = 0 as ε → 0, or (b) 0 < pε,12 → p0,12 = 0 as ε → 0, and pε,21 = 0, for ε ∈ (0, 1]. The condition V1 implies that vε = ∞, ε ∈ (0, 1]. The role of time compression factor is played by the function wε, ε ∈ (0, 1]. Note that wε = p−1 ε,21, ε ∈ (0, 1], if condition V1 (a) is satisfied, while wε = −1 pε,12, ε ∈ (0, 1], if condition V1 (b) is satisfied. Let us investigate the asymptotics of the probabilities Pε,i j (tε, A) for times 0 ≤ tε → ∞ as ε → 0 satisfying the following asymptotic relation: tε /wε → t ∈ [0, ∞] as ε → 0.

(9.1)

It is easy to see that the conditions V1 (a) and V1 (b) are in a sense stronger forms of the conditions S0 and S∞ , respectively. Therefore, it is expected that ergodic theorems for super-singularly perturbed alternating regenerative processes will take a form similar to that presented for singularly perturbed alternating processes in the short time ergodic Theorems 7.2–7.6. We also include in the class of super-singularly perturbed alternating regenerative processes the so-called absolutely singular perturbed alternating regenerative processes. This is the case, where additionally to O2 –R2 , the following condition is satisfied: V2 : pε,12, pε,21 = 0, for ε ∈ [0, 1].

9.1 Super-singularly perturbed ARP

165

9.1.2 Super-Long Time Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes In this subsection, we study the asymptotic behaviour of the probabilities Pε,i j (tε , A) for times 0 ≤ tε → ∞ as ε → 0, satisfying the following relation: tε /wε → ∞ as ε → 0. The corresponding limiting probabilities take the following form:  π (A) for A ∈ BZ, j = 1, i ∈ X, π0,(0)j (A) = 0,1 0 for A ∈ BZ, j = 2, i ∈ X, and π0,(∞) j (A) =



0 for A ∈ BZ, j = 1, i ∈ X, π0,2 (A) for A ∈ BZ, j = 2, i ∈ X.

(9.2)

(9.3)

(9.4)

The following theorems take place. Theorem 9.1 Let the conditions O2 , P¯ 2 , Q2 , R2 , and V1 (a) be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → ∞ as ε → 0, (9.5) Pε,i j (tε, A) → π0,(0)j (A) as ε → 0. Theorem 9.2 Let the conditions O2 , P¯ 2 , Q2 , R2 , and V1 (b) be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → ∞ as ε → 0, (9.6) Pε,i j (tε, A) → π0,(∞) j (A) as ε → 0. Proof Let us prove Theorem 9.1. In this case, the asymptotic behaviour of the probabilities Pε,11 (tε, A) is given by Theorem 2.1. Indeed, if ηε (0) = 1, then the condition V1 (a) implies that the process ξε (t), t ≥ 0, coincides with the process ξε,1 (t), t ≥ 0, and the process ηε (t) = 1, t ≥ 0. The conditions O2 , P¯ 2 , Q2 , R2 , and V1 (a) obviously imply that the conditions O1 , P1 , Q1 , and R1 are satisfied for the regenerative processes ξε,1 (t), t ≥ 0, with the regenerative times ζε,1,n . Moreover, in this case the corresponding stopping probabilities qε,1 ≡ 0. Thus, Theorem 2.1 can be applied. This gives the following relation, for any A ∈ Γ, and any 0 ≤ tε → ∞ as ε → 0: Pε,11 (tε, A) → π0,1 (A) as ε → 0.

(9.7)

Also, for any A ∈ Γ and t ≥ 0, Pε,12 (t, A) = 0, for ε ∈ (0, 1].

(9.8)

166

9 Ergodic theorems for super-singularly perturbed ARP

According to Lemma 6.1, if ηε (0) = 2, then F˜ε,2 (·) = P2 {wε−1 ζ˜ε,1 ≤ ·} ⇒ F˜0,2 (·) as ε → 0,

(9.9)

where F˜0,2 (u) = 1 − e−u/e0,2 , u ≥ 0, is an exponential distribution function with −1 . parameter e0,2 The probabilities Pε,11 (tε, A) and Pε,21 (tε, A) are connected by the following renewal type relation: ∫ tε Pε,21 (tε, A) = Pε,11 (tε − s, A)P2 { ζ˜ε,1 ∈ ds} 0 ∫ ∞ = Pε,11 (tε − swε, A)P2 {wε−1 ζ˜ε,1 ∈ ds}, (9.10) 0

where the function Pε,11 (tε − swε, A) is defined as 0 for tε − swε < 0. Take an arbitrary sε → s ∈ [0, ∞) as ε → 0. Obviously, (tε − sε wε )/wε = tε /wε − sε → ∞ as ε → 0. In this case, according to the relation (9.7), the following asymptotic relation holds, for A ∈ Γ and s ∈ [0, ∞): Pε,11 (tε − sε wε, A) → π0,1 (A) as ε → 0.

(9.11)

The relation (9.11) implies that, for A ∈ Γ and s ∈ [0, ∞), us

Pε,11 (tε − ·wε, A) −→ π0,1 (A) as ε → 0.

(9.12)

The relations (9.9) and (9.11) imply, by Lemma B.2, that the following relation takes place, for A ∈ Γ: ∫ ∞ −1 −s/e0,2 Pε,21 (tε, A) → π0,1 (A)e0,2 e ds = π0,1 (A) as ε → 0. (9.13) 0

According to Lemma 3.4, the phase space Z ∈ Γ. Also, π0,1 (Z) = 1. Thus, the relation (9.13) implies that the following relation holds, for A ∈ Γ: Pε,22 (tε, A) ≤ Pε,22 (tε, Z) = 1 − Pε,21 (tε, Z) → 1 − π0,1 (Z) = 0 as ε → 0. The proof of Theorem 9.1 is complete. The proof of Theorem 9.2 is similar.

(9.14)



9.1 Super-singularly perturbed ARP

167

9.1.3 Long Time Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes In this subsection, we study the asymptotic behaviour of the probabilities Pε,i j (tε , A) for times 0 ≤ tε → ∞ as ε → 0, satisfying the following relation: tε /wε → t ∈ (0, ∞) as ε → 0.

(9.15)

The corresponding limiting stationary probabilities take that following form, for t ∈ (0, ∞): ⎧ ⎪ ⎪ ⎪ ⎨ ⎪

π0,1 (A) 0 (0) π0,i j (t, A) = ⎪ (1 − e−t/e0,2 )π (A) 0,1 ⎪ ⎪ ⎪ e−t/e0,2 π0,2 (A) ⎩

A ∈ BZ, j A ∈ BZ, j A ∈ BZ, j A ∈ BZ, j

= 1, i = 2, i = 1, i = 2, i

= 1, = 1, = 2, = 2,

(9.16)

⎧ e−t/e0,1 π0,1 (A) for A ∈ BZ, j ⎪ ⎪ ⎪ ⎨ ⎪ (1 − e−t/e0,1 )π0,2 (A) for A ∈ BZ, j (∞) π0,i j (t, A) = ⎪ 0 for A ∈ BZ, j ⎪ ⎪ ⎪ for A ∈ BZ, j π0,2 (A) ⎩

= 1, i = 2, i = 1, i = 2, i

= 1, = 1, = 2, = 2.

(9.17)

for for for for

and

The following theorems take place. Theorem 9.3 Let the conditions O2 , P¯ 2 , Q2 , R2 , and V1 (a) be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → t ∈ (0, ∞) as ε → 0, (9.18) Pε,i j (tε, A) → πi(0) j (t, A) as ε → 0. Theorem 9.4 Let the conditions O2 , P¯ 2 , Q2 , R2 , and V1 (b) be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → t ∈ (0, ∞) as ε → 0, (∞) (9.19) Pε,i j (tε, A) → π0,i j (t, A) as ε → 0. Proof Let us prove Theorem 9.3. If the condition V1 (a) holds, then the relations (9.7) and (9.8) given in the proof of Theorem 9.1 describe the asymptotics of the probabilities Pε,1j (tε, A), j = 1, 2. Let us consider again the weak convergence relation (9.9) and the renewal type relation (9.10) connecting probabilities Pε,11 (tε, A) and Pε,21 (tε, A). Take an arbitrary sε → s ∈ [0, ∞) as ε → 0. Obviously, (tε − sε wε )/wε = tε /wε − sε → t − s as ε → 0, and, thus, tε − sε wε → ∞ as ε → 0, for 0 ≤ s < t, and tε − sε wε → −∞ as ε → 0, for s > t. Therefore, according to the relation (9.7) and the definition of Pε,11 (tε − swε, A) = 0, for tε − swε < 0, the following asymptotic relation holds, for A ∈ Γ and s  t: Pε,11 (tε − sε wε, A) → π0,1 (A)I(0 ≤ s < t) as ε → 0.

(9.20)

168

9 Ergodic theorems for super-singularly perturbed ARP

The relation (9.20) implies that, for A ∈ Γ and s  t, us

Pε,11 (tε − ·wε, A) −→ π0,1 (A)I(0 ≤ s < t) as ε → 0.

(9.21)

Note that the locally uniform convergence of the functions Pε,11 (tε − ·wε, A) as ε → 0 is not guaranteed for s = t. However, the limiting distribution in the relation (9.9) is exponential and, thus, has no atom at any point t > 0. Therefore, the relations (9.9) and (9.21) imply, by Lemma B.2, that the following relation takes place: for A ∈ Γ and t ∈ (0, ∞), ∫ ∞ −1 −s/e0,2 π0,1 (A)I(0 ≤ s < t)e0,2 e ds Pε,21 (tε, A) → 0

= (1 − e−t/e0,2 )π0,1 (A) as ε → 0.

(9.22)

It remains to carry out an asymptotic analysis of the asymptotic behaviour of probabilities Pε,22 (tε, A). Let us introduce random variables: με,2,n = κε,2,n I(ηε,2,n = 2), n = 1, 2, . . . .

(9.23)

Let now consider random sequence of stochastic triplets: ξ¯ε,2,n = ξε,2,n (t), t ≥ 0 , κε,2,n, με,2,n , n = 1, 2, . . . ,

(9.24)

a regenerative process: ξε,2 (t) = ξε,2,n (t − ζε,2,n−1 ), for t ∈ [ζε,2,n−1, ζε,2,n ), n = 1, 2, . . . ,

(9.25)

with regeneration times: ζε,2,n = κε,2,1 + · · · + κε,2,n, n = 1, 2, . . . , ζε,2,0 = 0,

(9.26)

and a random lifetime: με,2,+ = ζε,2,νε,2 ,

(9.27)

where νε,2 = min(n ≥ 1 : με,2,n < κε,2,n ) = min(n ≥ 1 : ηε,2,n = 1).

(9.28)

Let us also denote, for A ∈ Γ, t ≥ 0, Pε,2,+ (t, A) = P2 {ξε,2 (t) ∈ A, με,2,+ > t}.

(9.29)

In this case, the distribution functions: F¯ε,2 (t) = P{κε,2,1 ≤ t}, t ≥ 0, and

(9.30)

9.1 Super-singularly perturbed ARP

Fε,2,+ (t) = P{κε,2,1 ≤ t, με,2,1 ≥ κε,2,1 } = P{κε,2,1 ≤ t, ηε,2,1 = 2}, t ≥ 0.

169

(9.31)

Also, the stopping probability: qε,2,+ = P{με,2,1 < κε,2,1 } = P{ηε,2,1 = 1} = pε,21,

(9.32)

e¯ε,2 = E κε,2,1 = eε,21 + eε,22,

(9.33)

eε,2,+ = E κε,2,1 I(με,2,1 ≥ κε,2,1 ) = E κε,2,1 I(ηε,2,1 = 2) = eε,22 .

(9.34)

and the expectations: and

It is easy to see that, for every A ∈ BZ, t ≥ 0, Pε,22 (t, A) = P2 {ξε (t) ∈ A, ζ˜ε,1 > t} = P{ξε,2 (t) ∈ A, με,2,+ > t} = Pε,2,+ (t, A).

(9.35)

The conditions O2 , P¯ 2 , Q2 , R2 , and V1 (a) and imply that the conditions O1 , P1 , Q1 , and R1 are satisfied. Thus, the conditions of Theorem 9.3 imply that all conditions of Theorem 2.3 hold for the regenerative processes ξε,2 (t), t ≥ 0, with regenerative times ζε,2,n, n = 1, 2, . . ., and random lifetimes με,2,+ . Therefore, the following relation holds, for A ∈ Γ and any tε → t ∈ (0, ∞) as ε → 0, (9.36) Pε,22 (tε, A) = Pε,2,+ (tε, A) → e−t/e0,2 π0,2 (A) as ε → 0. It is worth noting that this proposition and its proof are similar to Lemma 6.5 and its proof. The proof of Theorem 9.3 is complete. The proof of Theorem 9.4 is similar. 

9.1.4 Short Time Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes In this subsection, we study the asymptotic behaviour of the probabilities Pε,i j (tε , A) for times 0 ≤ tε → ∞ as ε → 0, satisfying the following relation: tε /wε → 0 as ε → 0.

(9.37)

170

9 Ergodic theorems for super-singularly perturbed ARP

The corresponding limiting probabilities are the same for both cases, where condition V1 (a) or V1 (b) is satisfied. They take the following form:  π (A) for A ∈ BZ, j = i, i ∈ X, (9.38) π¯0,i j (A) = 0,i 0 for A ∈ BZ, j  i, i ∈ X. The following theorem takes place. Theorem 9.5 Let the conditions O2 , P¯ 2 , Q2 , R2 , and V1 be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → 0 as ε → 0, Pε,i j (tε, A) → π¯0,i j (A) as ε → 0.

(9.39)

Proof First, suppose that the condition V1 (a) is satisfied. In this case, the relations (9.7) and (9.8) given in the proof of Theorem 9.1 describe the asymptotics of the probabilities Pε,1j (tε, A), j = 1, 2. It is easy to see that, for every t ≥ 0, F˜ε,2 (t) = P2 { ζ˜ε,1 ≤ t} = P{με,2,+ ≤ t}

(9.40)

and, for every A ∈ BZ, t ≥ 0, P2 {ξε (t) ∈ A, ηε (t) = 2, ζ˜ε,1 > t} = P{ξε,2 (t) ∈ A, με,2,+ > t}.

(9.41)

d According to the relation (9.9), if ηε (0) = 2, the random variables, wε−1 ζ˜ε,1 −→ e0,2 ζ as ε → 0, where ζ is a random variable exponentially distributed with parameter 1. Since, we assumed that tε /wε → 0 as ε → 0, the above convergence in distribution relation and the relation (9.40) imply that,

P{με,2,+ > tε } = P2 { ζ˜ε,1 > tε }

= P2 {wε−1 ζ˜ε,1 > tε wε−1 } → 1 as ε → 0.

(9.42)

The relations (9.41) and (9.42) imply that P2 {ξε (tε ) ∈ A, ηε (tε ) = 2} − P2 {ξε (tε ) ∈ A, ηε (tε ) = 2, ζ˜ε,1 > tε }

≤ P2 { ζ˜ε,1 ≤ tε } → 0 as ε → 0,

(9.43)

and, similarly, P{ξε,2 (tε ) ∈ A} − P{ξε,2 (tε ) ∈ A, με,2,+ > tε }

≤ P{με,2,+ ≤ tε } → 0 as ε → 0.

(9.44)

These relations and Theorem 2.1, which can be applied to the regenerative processes ξε,2 (t), imply that, for every A ∈ Γ,

9.2 Super-singularly perturbed ARP compressed in time

171

lim P22 (tε, A) = lim P2 {ξε (tε ) ∈ A, ηε (tε ) = 2}

ε→0

ε→0

= lim P2 {ξε (tε ) ∈ A, ηε (tε ) = 2, ζ˜ε,1 > tε } ε→0

= lim P{ξε,2 (tε ) ∈ A, με,2,+ > tε } ε→0

= lim P{ξε,2 (tε ) ∈ A} = π0,2 (A). ε→0

(9.45)

If ηε (0) = 2, then, for every t > 0, event {ηε (t) = 1} ⊆ { ζ˜ε,1 ≤ t}. Thus, for every A ∈ Γ, P21 (tε, A) = P2 {ξε (tε ) ∈ A, ηε (tε ) = 1}

≤ P2 { ζ˜ε,1 ≤ tε } → 0 as ε → 0.

(9.46)

The proof for the case, where condition V1 (b) is satisfied, is absolutely similar to the above proof, due to the symmetry of the conditions O2 , P¯ 2 , Q2 , R2 , and V1 (a) and (b) with respect to indices i, j ∈ X.  As mentioned above, we also include absolutely singular perturbed alternating regenerative processes in the class of super-singularly perturbed alternating regenerative processes. This is the extremal case, where condition V2 is satisfied. In this case, the process ξε (t), t ≥ 0, coincides with the process ξε,i (t), t ≥ 0, and the process ηε (t) = i, t ≥ 0, if ηε (0) = i, for i = 1, 2. In this case, the asymptotics of the probabilities Pε,ii (tε, A) is given by Theorem 2.1. Also, the probabilities Pε,12 (t, A), Pε,21 (t, A) = 0, t ≥ 0, for all ε ∈ (0, 1]. The above remarks can be summarised in following theorem. Theorem 9.6 Let the conditions O2 , P¯ 2 , Q2 , R2 , and V2 be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0, Pε,i j (tε, A) → π¯0,i j (A) as ε → 0.

(9.47)

9.2 Ergodic Theorems for Super-Singularly Perturbed Alternating Regenerative Processes Compressed in Time In this section, we present additional ergodic theorems for super-singularly perturbed alternating regenerative processes with degenerating regeneration times.

172

9 Ergodic theorems for super-singularly perturbed ARP

9.2.1 Super-Singularly Perturbed Alternating Regenerative Processes with Degenerating Regeneration Times In this subsection, we present ergodic theorems for super-singularly perturbed alternating regenerative processes for the case where either the distribution functions F0,1j (·), j ∈ Y0,1 , or F0,2j (·), j ∈ Y0,2 , coincide with the distribution function F0 (u) = I(u ≥ 0) that has a unit jump in point 0. Theorems 9.7 and 9.8 and Lemma 9.1 and their proofs are similar, respectively, to Theorems 8.3 and 8.4 and Lemma 8.1 and their proofs. This allows us to omit the proofs of Theorems 9.7 and 9.8 and Lemma 9.1. We assume that the condition V1 is satisfied. In this case, the condition P¯ 2 can be replaced by one of the conditions P¯ 2 or P¯ 2. The following two super-long/long time ergodic theorems take place. Theorem 9.7 Let the conditions O2 , P¯ 2 , Q2 , R2 , and V1 (a) be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → t ∈ (0, ∞] as ε → 0, (9.48) Pε,i j (tε, A) → πi(0) j (t, A) as ε → 0. Theorem 9.8 Let the conditions O2 , P¯ 2, Q2 , R2 , and V1 (b) be satisfied. Then, for every A ∈ Γ, i, j ∈ X, and any 0 ≤ tε → ∞ as ε → 0 such that tε /wε → t ∈ (0, ∞] as ε → 0, (∞) (9.49) Pε,i j (tε, A) → π0,i j (t, A) as ε → 0. Remark 9.1 It should be noted that a theorem similar to Theorem 9.5 does not take place for the cases where the condition P¯ 2 is replaced by the condition P¯ 2 or P¯ 2. This is for reasons similar to those indicated in the Remark 8.4. These conditions are not enough to to get a relation similar to (9.42).

9.2.2 Compression in Time for Super-Singularly Perturbed Alternating Regenerative Processes The time compression procedure for singularly perturbed alternating regenerative processes is similar to that described in Sect. 8.2 for singularly perturbed alternating regenerative processes. Let us assume that the condition V1 is satisfied. ˆ 2, R ˆ 2 , and Xγ (formulated Also, we assume that the conditions O2 , Pˆ¯ 2 , Q in Sect. 8.2) are satisfied for the perturbed alternating regenerative processes (ξε (t), ηε (t)) with the regenerative times ζε,n . It is also useful to note that, according to the relations (8.39) and (8.42), the corresponding limiting stationary probabilities:

9.3 Generalisations and classification of ergodic theorems for perturbed ARP (γ)

π0,i (A) = =

1

∫

(γ)

e0,i

1 e0,i

∫

∞

0 ∞

173

(γ)

q0,i (s, A)m(ds) q0,i (s, A)m(ds) = π0,i (A),

(9.50)

0

do not depend on the parameter γ, in the cases γ ∈ [0, ∞), i = 1, A ∈ BZ and γ ∈ (0, ∞], i = 2, A ∈ BZ . The following lemma takes place. ˆ 2, R ˆ 2 , V1 , Xγ (for some γ ∈ [0, ∞]) be Lemma 9.1 Let the conditions O2 , Pˆ¯ 2 , Q satisfied for the alternating regenerative processes (ξε (t), ηε (t)). Then the following conditions are satisfied for the alternating regenerative processes (ξε,uε (t), ηε,uε (t)) compressed in time by the factor uε given by the relation (8.31): (i) If γ ∈ (0, ∞), the conditions O2 , P¯ 2 , Q2 , and R2 are satisfied. (ii) If γ = 0, the conditions O2 , P¯ 2 , Q2 , and R2 are satisfied. (iii) If γ = ∞, the conditions O2 , P¯ , Q2 , and R2 are satisfied. 2

(iv) The asymptotic relations (8.35), (8.36), (8.38), and (8.41) play the roles of the asymptotic relations appearing in the above conditions. The function uε is given by relation (8.31). The corresponding limiting quantities and sets appearing in these conditions and related relations are given by the relations (8.34)–(8.40) and (9.50). Remark 9.2 Lemma 9.1 allows us to apply to the alternating regenerative processes ˆ 2, (ξε,uε (t), ηε,uε (t)) Theorems 9.1–9.4, 9.7, and 9.8, if the conditions O2 , Pˆ¯ 2 , Q ˆ R2 , V1 (a) or V1 (b), and Xγ are satisfied, and tε /wε → t ∈ (0, ∞]. In particular, Theorem 9.1 can be applied, if the condition V1 (a) is satisfied, γ ∈ (0, ∞) and t = ∞, Theorem 9.2, if the condition V1 (b) is satisfied, γ ∈ (0, ∞) and t = ∞, Theorem 9.3, if the condition V1 (a) is satisfied, γ ∈ (0, ∞) and t ∈ (0, ∞), Theorem 9.4, if the condition V1 (b) is satisfied, γ ∈ (0, ∞), and t ∈ (0, ∞), Theorem 9.7, if the condition V1 (a) is satisfied, γ = 0, and t ∈ (0, ∞], Theorem 9.8, and if the condition V1 (b) is satisfied, γ = ∞ and t ∈ (0, ∞].

9.3 Generalisations and Classification of Ergodic Theorems for Perturbed Alternating Regenerative Processes In this section, we discuss some generalisations of ergodic theorems for perturbed alternating regenerative processes, based on possibilities to weaken some model assumptions or perturbation conditions. Also, we present the classification of ergodic theorems for perturbed alternating regenerative processes, which summarises the results presented in Chaps. 3–9.

174

9 Ergodic theorems for super-singularly perturbed ARP

9.3.1 Generalisations of Ergodic Theorems for Perturbed Alternating Regenerative Processes 9.3.1.1 Ergodic Theorems for Perturbed Alternating Regenerative Processes with Arbitrary Initial Distributions of the Modulating Component. Let us comment on ergodic theorems for perturbed alternating regenerative processes with general perturbed initial distributions p¯ε = pε,1, pε,2 for modulating semi-Markov processes. In this case, the objects of study are the probabilities: Pε, p¯ ε , j (t, A) = P p¯ ε {ξε (t) ∈ A, ηε (t) = j} = pε,1 Pε,1j (t, A) + pε,2 Pε,2j (t, A), j = 1, 2, t ≥ 0,

(9.51)

where the probabilities pε,i ≥ 0, i = 1, 2, pε,1 + pε,2 = 1, for ε ∈ (0, 1]. In models where the corresponding limits for the probabilities Pε,i j (tε, A) do not depend on the initial state i, for example, for regularly perturbed alternating regenerative processes, the probabilities Pε, p¯ ε , j (tε, A) converge to the same limits for any initial distributions p¯ε = pε,1, pε,2 . However, in models where the corresponding limits for the probabilities Pε,i j (tε, A) may depend on the initial state i, for example, for some singularly or super-singularly perturbed alternating regenerative processes, the probabilities Pε, p¯ ε , j (t, A) converge to some limits, under the following additional condition of asymptotic stability of initial distributions: W: pε,i → p0,i as ε → 0, for i ∈ X. If, for example, the condition W is satisfied, and for i, j ∈ X, A ∈ Γ and any 0 ≤ tε → ∞ such that tε /vε → t ∈ [0, ∞] as ε → 0, Pε,i j (tε, A) → π0,i j (t, A) as ε → 0,

(9.52)

then Pε, p¯ ε , j (tε, A) → π0, p¯0, j (t, A) = p0,1 π0,1j (t, A) + p0,2 π0,2j (t, A) as ε → 0.

(9.53)

9.3.1.2 Alternating Regenerative Processes with Transition Periods. Ergodic theorems for perturbed alternating regenerative processes can be generalised to such processes with transition periods. In this case, the model assumption (J) formulated in Sect. 3.1.1 is assumed to fulfil only for n ≥ 2. The alternating regenerative process (ξε (t), ηε (t)), t ≥ 0, has the transition period [0, ζε,1 ). The shifted process (ξε(1) (t), ηε(1) (t)) = (ξε (ζε,1 + t), ηε (ζε,1 + t)) ≥ 0 is a usual alternating regenerative process. All quantities appearing in the conditions O2 –R2 , etc., the renewal type equations (3.8), and the relations (3.7) and (3.9) should be, in this case, defined using the shifted sequence of triplets ξ¯ε,i,2 = ξε,i,2 (t), t ≥ 0 , κε,i,2, ηε,i,2 , i ∈ X. It is also natural to index the above-mentioned quantities by the upper index (1) , for example, to use the

9.3 Generalisations and classification of ergodic theorems for perturbed ARP

175

(1) (1) (1) (1) notation Pε,i, j (t, A) = Pi {ξε (t) ∈ A, ηε (t) = j}, etc. The probabilities Pε,i j (t, A) satisfy the system of renewal type equations (3.8). Theorems 4.1–9.8 present, in this case, the corresponding ergodic relations for these probabilities. Moreover, it is possible to generalise the model and assume that the random  = {1, . . . , m},  which differs from the phase space variable ηε has a phase space X  is X = {1, 2} and the family of triplets ξ¯ε,1,i = ξε,1,i (t), t ≥ 0 , κε,1,i, ηε,1,i , i ∈ X,  ¯ used for the transition period. Here, ξε,1,i = ξε,1,i (t), t ≥ 0 , i ∈ X, are measurable  are non-negative random variables, processes with the phase space Z, κε,1,i, i ∈ X,  and ηε,1,i, , i ∈ X, are random variables taking values in the space X. As above, it is assumed that the random variable ηε , triplets ξ¯ε,1,i = ξε,1,i (t), t ≥  and ξ¯ε,n,i = ξε,n,i (t), t ≥ 0 , κε,n,i, ηε,n,i , n ≥ 2, i ∈ X, 0 , κε,1,i, ηε,1,i , i ∈ X, are mutually independents and the stochastic triplets ξ¯ε,n,i = ξε,n,i (t), t ≥ 0 , κε,n,i, ηε,n,i , n ≥ 2, are probabilistic copies of the triplet ξ¯ε,2,i = ξε,2,i (t), t ≥ 0 , κε,2,i, ηε,2,i , for every i ∈ X (in the sense of holding the model assumption (J)). The corresponding alternating regenerative process (ξε (t), ηε (t)) with transition period [0, ζε,1 ) is defined using the same recurrent relations (3.1)–(3.3).  j ∈ X, t ≥ 0, and ε ∈ (0, 1], Let us denote, for i ∈ X,

and

pε,i j = P{ηε,i,1 = j},

(9.54)

Q ε,i j (t) = P{κε,i,1 ≤ t, ηε,i,1 = j}.

(9.55)

 and ε ∈ (0, 1], According to the relations (9.54) and (9.55), for i ∈ X

pε,i j ≥ 0, j ∈ X, pε,i j = 1,

(9.56)

j ∈X

and

lim Q ε,i j (t) = Q ε,i j (∞) = pε,i j as t → ∞, for j ∈ X.

t→∞

(9.57)

The following condition is similar to the condition P 1 :  j ∈ X, (b) Q ε,i j (·) ⇒ Q 0,i j (·) as ε → 0, P 2 : (a) pε,i j → p0,i j as ε → 0, for i ∈ X,  j ∈ X, where Q 0,i j (·) are proper or improper distribution functions such for i ∈ X,  j ∈ X. that Q 0,i j (∞) = p0,i j , i ∈ X, The relations (9.56)–(9.57) and the condition P 2 imply that the following relation  holds, for i ∈ X:

p0,i j = 1. (9.58) p0,i j ≥ 0, j ∈ X, j ∈X

The case, where some or even both distribution functions, Q 0,i (·) = I(u ≥ 0) p0,i j , is also admissible in the condition P 2 (b).  Note also that the condition P 2 implies that, for i ∈ X,

176

9 Ergodic theorems for super-singularly perturbed ARP

Fε,i (·) =



Q ε,i j (·)

j ∈X

⇒



Q 0,i j (·) = F0,i (·) as ε → 0.

(9.59)

j ∈X

The condition P 2 and the relation (9.58) imply that F0,i (·) is a proper distribution  function, for i ∈ X. The corresponding ergodic relations for the probabilities Pε,i j (tε, A) = Pi {ξε (t) ∈ A, ηε (t) = j} take the form similar to the asymptotic relation (9.53). If, for example, for i, j ∈ X, A ∈ Γ and any 0 ≤ tε → ∞ such that tε /vε → t ∈ [0, ∞] as ε → 0, (1) Pε,i j (tε , A) → π0,i j (t, A) as ε → 0,

(9.60)

 j ∈ X, then, for i ∈ X, Pε,i j (tε, A) → π0, p0, i , j (t, A) = p0,i1 π0,1j (t, A) + p0,i2 π0,2j (t, A) as ε → 0,

(9.61)

 where p0,i = p0,i1, p0,i2 , i ∈ X. (1) Indeed, let us define Pε,i j (tε − s, A) = 0 for s > tε . Note that (tε − sε )/vε → t as ε → 0, for any 0 ≤ sε → s ∈ [0, ∞) as ε → 0. The relation (9.60) implies that, for i, j ∈ X, A ∈ Γ and any 0 ≤ sε → s ∈ [0, ∞) as ε → 0, (β) (1) (9.62) Pε,i j (tε − sε , A) → π0,i j (t, A) as ε → 0.  2 , P 2 , the relations (9.59) and (9.62), and Lemma B.2 Finally, the conditions O  imply that, for i ∈ X, j ∈ X, A ∈ Γ and any 0 ≤ tε → ∞ such that tε /vε → t ∈ [0, ∞] as ε → 0, Pε,i j (tε, A) = Pi {ξε (tε ) ∈ A, ηε,tε = j, ζε,1 > tε } 2 ∫ ∞

(1) + Pε,k (t − s, A)Q ε,ik (ds) j ε k=1

→

2

0

π0,k j (t, A) p0,ik as ε → 0,

(9.63)

k=1

since Pi {ξε (tε ) ∈ A, ηε,tε = j, ζε,1 > tε } ≤ 1 − Fε,i (tε ) → 0 as ε → 0.

(9.64)

9.3.1.3 Ergodic Theorem for Perturbed Regenerative and Alternating Regenerative Processes, Which Admit Instant Transitions. This is the model, in which, respectively, condition O1 (a) or O2 (b) is weakened.

9.3 Generalisations and classification of ergodic theorems for perturbed ARP

177

For the model of perturbed regenerative processes, condition O1 (a) can be replaced by the weaker assumption that P{κε,1 = 0} < 1, for ε ∈ (0, 1]. In this case, let us define stopping moments: ιε,n = min(r > ιε,n−1, κε,r > 0), n = 1, 2, . . . , ιε,0 = 0,

(9.65)

and new stochastic triplets, for n = 1, 2, . . ., ◦ ◦ ◦ ◦ ξ¯ε,n = ξε,n (t) = ξε,ιε, n (t), t ≥ 0 , κε,n = κε,ιε, n , ηε,n = ηε,ιε, n .

(9.66)

The above triplets satisfy assumptions (A)–(E) formulated in Sect. 2.1.1.1. ◦ > 0} = 1. Also, according to the relation (9.65), P{κε,1 Finite-dimensional distributions of the initial triplets ξ¯ε,n = ξε,n (t), t ≥ ◦ = ξ ◦ (t), t ≥ 0 , κ ◦ , η ◦ , n = 0 , κε,n, ηε,n , n = 1, 2, . . ., and the new triplets ξ¯ε,n ε,n ε,n ε,n 1, 2, . . ., are connected by the following simple relation, which holds, for any 0 ≤ t1 ≤ · · · ≤ tl, A1, . . . , Al ∈ BZ, l = 1, 2, . . . , u ≥ 0, ı = 1, 2, ◦ ◦ ◦ P{ξε,1 (tr ) ∈ Ar , r = 1, . . . , l, κε,1 ≤ u, ηε,1 = ı}

= P{ξε,1 (tr ) ∈ Ar , r = 1, . . . , l, κε,1 ≤ u, ηε,1 = ı / κε,1 > 0}.

(9.67)

It is easy to see that the regenerative process ξε (t), t ≥ 0, with the regenerative lifetime με , constructed with the use of the initial triplets ξ¯ε,n = ξε,n (t), t ≥ 0 , κε,ιε, n , ηε,ιε, n , n = 1, 2, . . ., and the relations (2.1)–(2.4), and the regenerative process ξε◦ (t), t ≥ 0, with the regenerative lifetime μ◦ε , constructed with the use of the above new stochastic triplets and the relations similar to (2.1)–(2.4), coincide P{ξε (t) = ξε◦ (t), t ≥ 0, με = μ◦ε } = 1.

(9.68)

◦ Let us assume that the conditions O1 –R1 are imposed on the triplets ξ¯ε,n = ◦ ◦ ≥ 0 , κε,n, ηε,n , n = 1, 2, . . .. The above remarks make it possible to transfer the perturbation conditions O1 –R1 on the triplets ξ¯ε,n = ξε,n (t), t ≥ 0 , κε,n, ηε,n , n = 1, 2, . . .. Owing to the relation (9.68), this makes it possible to reformulate the corresponding ergodic theorems for the perturbed regenerative processes ξε◦ (t), with the regenerative lifetimes μ◦ε in terms of the regenerative processes ξε (t) with the regenerative lifetimes με . For the model of perturbed alternating regenerative processes, the condition O2 (b) can be replaced by the weaker assumption that P{κε,i,1 = 0} < 1, i ∈ X, for ε ∈ [0, 1], in the case where the additional model assumption: P{κε,i,1 = 0, ηε,i,1  i} = 0, i ∈ X, holds, for ε ∈ [0, 1]. In this case, the embedding into the model, where the condition P{κε,i,1 = 0} = 0, i ∈ X, for ε ∈ (0, 1], holds, can be implemented in a manner similar to that described above for perturbed regenerative processes. 9.3.1.4 Weaker Arithmetic Assumptions for Limiting Distributions of Return Times for Regularly Perturbed Alternating Regenerative Processes. Let us first consider the case of regularly perturbed standard alternating regenerative processes, where the condition T1 is assumed to be satisfied. ◦ (t), t ξε,n

178

9 Ergodic theorems for super-singularly perturbed ARP

The condition P2 (c) can be weakened in Theorem 4.1. It can be replaced by the weaker assumption that the distribution function F0,12 (·) or F0,21 (·) is non-arithmetic, while, respectively, F0,21 (·) or F0,12 (·) is an arbitrary distribution function concentrated on the interval [0, ∞) (including the case where it is F0 (u) = I(u ≥ 0)). In the proof of Theorem 4.1, the distribution functions Fˆε,11 (·) (generating the renewal equations (4.24)) weakly converge as ε → 0 to the distribution function Fˆ0,11 (·) = F0,12 (·) ∗ F0,21 (·) given by the relation (4.27). The above-mentioned weaker variant of the condition P2 (c) implies that the distribution function Fˆ0,11 (·) is non-arithmetic. Indeed, let us introduce the characteristic functions: ∫ ∞ ϕˆ0,11 (z) = eizu Fˆ0,11 (du), z ∈ R1, (9.69) 0

∫

and ϕ0,i j (z) =

0

∞

eizu Q0,i j (du), z ∈ R1, i, j ∈ X.

(9.70)

According to relation (4.27), the characteristic function ϕˆ0,11 (z) is given by the following relation: (9.71) ϕˆ0,11 (z) = ϕ0,12 (z)ϕ0,21 (z), z ∈ R1 . According to Lemma A.1, the above assumption imposed on distribution functions F0,i j (·) imply that |ϕ0,12 (z)| < 1, z  0, or |ϕ0,21 (z)| < 1, z  0. Then, | ϕˆ0,11 (z)| ≤ |ϕ0,12 (z)||ϕ0,21 (z)| < 1, z  0,

(9.72)

and, thus, by Lemma A.1, the distribution function Fˆ0,11 (·) is non-arithmetic. Moreover, Theorems 2.1–2.3, used in the proof of Theorem 4.1, require only that the distribution function Fˆ0,11 (·) would be weakly non-arithmetic. Thus, condition P2 (c) can, in fact, be replaced by the assumption that the distribution function Fˆ0,11 (·) is weakly non-arithmetic. The distribution function Fˆ0,11 (t) is weakly non-arithmetic if the distribution function F0,12 (·) or F0,21 (·) is weakly non-arithmetic, while, respectively, F0,21 (·) or F0,12 (·) is F0 (u) = I(u ≥ 0). The distribution function Fˆ0,11 (·) can be weakly non-arithmetic even in the case where both distribution functions F0,12 (·), F0,21 (·) are arithmetic. Let us assume that the distribution functions F0,12 (·) and F0,21 (·) are arithmetic and have spans and shifts, respectively, h12, d1,2 and h21, d21 . The distribution function Fˆ0,11 (·) is non-arithmetic if the quotient h12 /h21 is an irrational number. If the quotient h12 /h21 = r12 /r21 is a rational number (given in the irreducible form), then the distribution function Fˆ0,11 (·) is arithmetic, with the span h12 /r12 = 21 h21 /r21 = h and the shift dh = d12 + d21 − [ d21 +d ]h. h If the shift dh = 0, the distribution function Fˆ0,11 (·) is also strictly arithmetic.

9.3 Generalisations and classification of ergodic theorems for perturbed ARP

179

However, if the shift dh ∈ (0, h), the distribution function Fˆ0,11 (·) is arithmetic but not strictly arithmetic, i.e. it is weakly non-arithmetic. Let us now consider the general case of regularly perturbed alternating regenerative processes, for which the condition T2 is satisfied. In this case, the condition P2 (c) can also be weakened in Theorem 4.2. It can be replaced by the above formulated weaker assumption that the distribution function F0,12 (·) or F0,21 (·) is non-arithmetic, while, respectively, F0,21 (·) or F0,12 (·) is an arbitrary distribution function concentrated on interval [0, ∞) (including the case, where it is F0 (u) = I(u ≥ 0)). In the proof of Theorem 4.2, the distribution functions Fˆε,11 (·) generating the renewal equations (4.52) weakly converge as ε → 0 to the distribution function Fˆ0,11 (·) given by relation (4.60). The above weakened condition implies that the distribution function Fˆ0,11 (·) is non-arithmetic. Indeed, let us introduce the characteristic function: ∫ ∞ ϕˆ0,11 (z) = eizu Fˆ0,11 (du), z ∈ R1 . (9.73) 0

According to the relation (4.62), this characteristic function is given by the following relation: ϕˆ0,11 (z) = ϕ0,11 (z)p0,11 +

ϕ0,12 (z)p0,12 ϕ0,21 (z)p0,21 , z ∈ R1 . 1 − ϕ0,22 (z)p0,22

(9.74)

Note that, according to the condition T2 , the probabilities p0,12, p0,21 > 0. If |ϕ0,12 (z)| < 1, z  0, or |ϕ0,21 (z)| < 1, z  0, then |ϕ0,12 (z)|p0,12 |ϕ0,21 (z)|p0,21 1 − |ϕ0,22 (z)|p0,22 |ϕ0,12 (z)|p0,12 |ϕ0,21 (z)|p0,21 ≤ p0,11 + 1 − p0,22 = p0,11 + |ϕ0,12 (z)||ϕ0,21 (z)|p0,12 < 1, z  0,

| ϕˆ0,11 (z)| ≤ |ϕ0,11 (z)|p0,11 +

(9.75)

and, thus, by Lemma A.1, the distribution function Fˆ0,11 (·) is non-arithmetic. Finally, let us consider the case of semi-regularly perturbed alternating regenerative processes, for which the condition T3 holds. The condition P2 (c) can be weakened in Theorems 4.3. The condition can be replaced by the weaker assumption that the distribution function F0,11 (·) is weakly non-arithmetic, while F0,2j (·), j ∈ Y1,2 , can be arbitrary distribution functions concentrated on the interval [0, ∞). Indeed, in the proof of Theorem 4.3, the distribution functions Fˆε,11 (·) generating the renewal equations (4.85) weakly converge as ε → 0 to the distribution function Fˆ0,11 (·) = F0,11 (·). Similarly, the condition P2 (c) can also be weakened in Theorem 4.4.

180

9 Ergodic theorems for super-singularly perturbed ARP

It can be replaced by the weaker assumption that the distribution function F0,22 (·) is weakly non-arithmetic, while F0,1j (·), j ∈ Y1,1 , can be arbitrary distribution functions concentrated on the interval [0, ∞). Finally it is worth noting that the proofs of Theorems 4.1–4.4 are based on the application of Theorems 2.1–2.3 to the corresponding perturbed embedded regenerative processes of the first type. These theorems only require that the limiting distribution function for their regeneration times be weakly non-arithmetic. The weak non-arithmetic assumption can be imposed directly on this limiting distribution function. This assumption can also be effective in cases where the characteristic function of the above limiting distribution function can be computed explicitly in a form that allows one to apply the necessary and sufficient criterion for a distribution function to be weakly non-arithmetic given in Lemma A.1.

9.3.2 Classification of Ergodic Theorems for Perturbed Alternating Regenerative Processes The forms of the corresponding ergodic relations and limiting probabilities given in the ergodic theorems presented in Chaps. 3–9 are essentially determined by two parameters. The first is the parameter β ∈ [0, ∞], which asymptotically balances the switching probabilities pε,12 and pε,21 with the following asymptotic relation: pε,12 → β ∈ [0, ∞] as ε → 0. pε,21

(9.76)

The second is the time scaling parameter t ∈ [0, ∞], which defines the asymptotic time zones for the time tε → ∞ as ε → 0, in the form of one of the two asymptotic relations: tε → t ∈ [0, ∞] as ε → 0, (9.77) vε −1 with the time compression factor: vε = p−1 ε,12 + pε,21 , or

tε → t ∈ [0, ∞] as ε → 0, wε

(9.78)

with the time compression factor: wε = (pε,12 + pε,21 )−1 . The variants of ergodic relations are presented in Theorems 4.1–9.8, which we split in groups as ergodic theorems for regularly perturbed alternating regenerative processes, and short, long, and super-long time ergodic theorems for singularly and super-singularly perturbed alternating regenerative processes. The classification of the respective individual ergodic theorems is summarised in the Tables 9.1, 9.2, and 9.3 (where the numbers of theorems, their conditions, the corresponding asymptotic time zones, and limiting probabilities are, respectively, given in columns 1, 2, 3, and 4).

9.3 Generalisations and classification of ergodic theorems for perturbed ARP T

Conditions

181

Asymptotic time zones Limiting probabilities

4.1 O2 , P2 , Q2 , R2 , T1 , β = 1

tε → ∞

π0,(1)j (A)

4.2 O2 , P2 , Q2 , R2 , T2 , β ∈ (0, ∞)

tε → ∞

π0, j (A)

4.3 O2 , P2 , Q2 , R2 , T3 , β = 0

tε → ∞

π0,(0)j (A)

4.4 O2 , P2 , Q2 , R2 , T3 , β = ∞

tε → ∞

π0,(∞)j (A)

5.1 O2 , P2 , Q2 , R2 , T4 (a), β ∈ [0, ∞)

tε → ∞

π0,(0)j (A)

5.2 O2 , P2 , Q2 , R2 , T4 , (b), β ∈ (0, ∞]

tε → ∞

π0,(∞)j (A)

(β)

Table 9.1 Classification of ergodic theorems: regular perturbations

T

Conditions

Asymptotic time zones

Lim. prob.

6.1 O2 , P¯ 2 , Q2 , R2 , U1 , Sβ , β ∈ [0, ∞]

vε ≺ t ε

π0, j (A)

6.2 O2 , P¯ 2 , Q2 , R2 , U1 , Sβ , β ∈ [0, ∞]

tε ∼ tvε , t ∈ (0, ∞)

π0, i j (t, A)

7.1 O2 , P¯ 2 , Q2 , R2 , U1 , Sβ , β ∈ (0, ∞)

t ε ≺ vε , t ε → ∞

π¯ 0, i j (A)

7.2 O2 , P¯ 2 , Q2 , R2 , U1 , S0

wε ≺ t ε ≺ vε

π0,(0)j (A)

7.3 O2 , P¯ 2 , Q2 , R2 , U1 , S∞

wε ≺ t ε ≺ vε

π0,(∞)j (A)

7.4 O2 , P¯ 2 , Q2 , R2 , U1 , S0

tε ∼ twε , t ∈ (0, ∞)

π 0,(0)i j (t, A)

7.5 O2 , P¯ 2 , Q2 , R2 , U1 , S∞

tε ∼ twε , t ∈ (0, ∞)

π 0,(∞) i j (t, A)

t ε ≺ wε , t ε → ∞

π¯ 0, i j (A)

8.1 O2 , P¯ 2 , Q2 , R2 , U1 , Sβ , β ∈ [0, ∞)

vε ≺ tε or tε ∼ tvε , t ∈ (0, ∞)

π0,(0)j (A)

8.2 O2 , P¯ 2 , Q2 , R2 , U1 , Sβ , β ∈ (0, ∞]

vε ≺ tε or tε ∼ tvε , t ∈ (0, ∞)

π0,(∞)j (A)

8.3 O2 , P¯ 2 , Q2 , R2 , U1 , S0

wε ≺ tε ≺ vε or tε ∼ twε , t ∈ (0, ∞)

π0,(0)j (A)

8.4 O2 , P¯ 2 , Q2 , R2 , U1 , S∞

wε ≺ tε ≺ vε or tε ∼ twε , t ∈ (0, ∞)

π0,(∞)j (A)

7.6 O2 , P¯ 2 , Q2 , R2 , U1 , S0 or S∞

(β)

(β)

Table 9.2 Classification of ergodic theorems: singular perturbations

It should be noted that the limiting probabilities, appearing in Theorems 4.1–9.8, (β) (β) (β) (0) have the forms π0, j (A) = ρ j (β)π0, j (A), π0,i j (t, A) = pi j (t)π0, j (A), π0,i j (t, A) = (∞) (∞) p(0) i j (t)π0, j (A), π 0,i j (t, A) = pi j (t)π0, j (A), and π¯ 0,i j (A). (β)

(∞) The coefficients ρ j (β), pi j (t), p(0) i j (t), and pi j (t) can be interpreted as, respectively, limiting stationary probabilities or transition probabilities for some limiting

182 T

9 Ergodic theorems for super-singularly perturbed ARP Conditions

Asymptotic time zones

Lim. probabilities

9.1 O2 , P¯ 2 , Q2 , R2 , V1 (a)

wε ≺ t ε

π0,(0)j (A)

9.2 O2 , P¯ 2 , Q2 , R2 , V1 (b)

wε ≺ t ε

π0,(∞)j (A)

9.3 O2 , P¯ 2 , Q2 , R2 , V1 (a)

tε ∼ twε , t ∈ (0, ∞)

π 0,(0)i j (t, A)

9.4 O2 , P¯ 2 , Q2 , R2 , V1 (b)

tε ∼ twε , t ∈ (0, ∞)

π 0,(∞) i j (t, A)

9.5 O2 , P¯ 2 , Q2 , R2 , V1

t ε ≺ wε , t ε → ∞

π¯ 0, i j (A)

9.6 O2 , P¯ 2 , Q2 , R2 , V2

tε → ∞

π¯ 0, i j (A)

9.7 O2 , P¯ 2 , Q2 , R2 , V1 (a)

wε ≺ tε or tε ∼ twε , t ∈ (0, ∞)

π0,(0)j (A)

9.8 O2 , P¯ 2 , Q2 , R2 , V1 (b)

wε ≺ tε or tε ∼ twε , t ∈ (0, ∞)

π0,(∞)j (A)

Table 9.3 Classification of ergodic theorems: super-singular perturbations

semi-Markov processes or Markov chains for the corresponding modulating semiMarkov processes, while π0, j (A) are the limiting stationary probabilities for the regenerative processes corresponding to different states of the above modulating semi-Markov processes. (β) (β) (0) It is worth noting that the limiting probabilities π0, j (A) and π0,i j (t, A), π0,i j (t, A),

(∞) π0,i j (t, A) possess some natural continuity properties as functions of the parameters β ∈ [0, ∞] and t ∈ [0, ∞]. (β) In particular, the limiting probabilities π0, j (A), which appear in Theorems 4.1– 4.4, 5.1, and 5.2, for regularly perturbed alternating regenerative processes, in Theorems 6.1, 7.2, 7.3, and 8.1–8.4, for singularly perturbed alternating regenerative processes, and in Theorems 9.1, 9.2, 9.7, and 9.8, for super-singularly perturbed alternating regenerative processes, are continuous functions of the parameter β ∈ [0, ∞]. (β) Analogously, the limiting probabilities π0,i j (t, A), which appear in Theorem 6.2 for singularly perturbed alternating regenerative processes, are continuous functions of the parameter (β, t) ∈ [0, ∞] × [0, ∞], except the points (0, 0) and (∞, 0). (0) (∞) Also, the limiting probabilities π0,i j (t, A) and π 0,i j (t, A), which appear in Theorems 7.4, 7.5, 9.3, and 9.4 for singularly and super-singularly perturbed alternating regenerative processes, are continuous functions of the parameter t ∈ [0, ∞]. (β) (β) The limits, π0,i j (0, A) = limt→0 π0,i j (t, A) = π0,i j (A), for β ∈ (0, ∞), while (0) (0) (0) (∞) (∞) π0,i j (0, A) = limt→0 π0,i j (t, A) = π0, j (A) and π0,i j (0, A) = limt→0 π0,i j (t, A) = (β)

(β)

(β)

π0,(∞) j (A). Also, the limit, π0,i j (∞, A) = limt→∞ π0,i j (t, A) = π0, j (A), for β ∈ [0, ∞]. Here, π0,i j (A) are the limiting probabilities appearing in Theorems 7.3, 7.6, 9.5, and 9.6. The above asymptotic relations have a natural explanation. In fact, there is some kind of “competition” between the rates with which the switching probabilities

9.3 Generalisations and classification of ergodic theorems for perturbed ARP

183

pε,12, pε,21 tend to zero and the time parameter tε tends to infinity, for singularly and super-singularly perturbed alternating regenerative processes. Probabilities pε,12 and pε,21 determine the “grade of singularity” for perturbed alternating regenerative processes. These processes become more singular if the parameter βε = pε,12 /pε,21 takes a small value close to 0 or a large value “close” to ∞. The time parameter t controls the “degree of ergodicity” of the perturbed alternating regenerative processes. Values of βε close to 0 or ∞, and smaller values of parameter t facilitate the convergence of the probabilities Pε,i j (tε, A) to the limiting probabilities π0,i j (A) = I( j = i)π0,i (A), typical for absolutely singular alternating regenerative processes (for which, switching of regimes is impossible). Moderate values of βε , asymptotically separated of 0 and ∞, and large values of t favour the manifestation of ergodic phenomena and convergence of the probabilities Pε,i j (tε, A) (β) to the limiting probabilities π0, j (A) = ρ j (β)π0, j (A), which are typical for regular alternating regenerative processes.

Part II

Ergodic Theorems for Perturbed Multi-Alternating Regenerative Processes

Chapter 10

Perturbed Multi-Alternating Regenerative Processes

In this chapter, we introduce a model of perturbed multi-alternating regenerative processes modulated by regularly or singularly perturbed finite semi-Markov processes (briefly called perturbed multi-alternating regenerative processes). Asymptotic recurrent algorithms of time-space aggregation of regeneration times based on the removal of virtual transitions and reduction of the phase space for modulating semiMarkov processes are described. This chapter consists of three sections. In Sect. 10.1, we introduce the model of perturbed multi-alternating regenerative processes and formulate the corresponding perturbation conditions. In Sect. 10.2, we describe asymptotic procedures of aggregation of regeneration times based on total and partial removing of virtual transitions for modulating semi-Markov processes for perturbed multi-alternating regenerative processes. The transformation of perturbation conditions is described for the procedure of total removing of virtual transitions, in Lemmas 10.4–10.10, and for the procedure of partial removing of virtual transitions, in Lemmas 10.11–10.18. In Sect. 10.3, we describe two asymptotic procedures of aggregation of regeneration times based on one-state reduction of phase space for modulating semi-Markov processes with totally and partially removed virtual transitions. The transformation of perturbation conditions is described for the first procedure, in Lemmas 10.19–10.24, and for the second one, in Lemmas 10.25–10.31.

10.1 Multi-Alternating Regenerative Processes In this section, we introduce the model of perturbed multi-alternating regenerative processes and formulate basic perturbation conditions.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0_10

187

188

10 Perturbed MARP

10.1.1 Definition of Multi-Alternating Regenerative Processes 10.1.1.1 Standard Multi-Alternating Regenerative Processes. Let Ωε, Fε, Pε

be, for every ε ∈ (0, 1], a probability space. We assume that all stochastic processes and random variables introduced below and indexed by parameter ε are defined on the probability space Ωε, Fε, Pε . Let X = {1, . . . , m} be a finite set. We assume that the following model assumptions are fulfilled for random processes and random variables defined below: (K) ξ¯ε,i,n = ξε,i,n (t), t ≥ 0 be, for every i ∈ X, n = 1, 2, . . ., a measurable stochastic process with a phase space Z. (L) κε,i,n, i ∈ X, n = 1, 2, . . . be non-negative random variables. (M) ηε and ηε,i,n, i ∈ X, n = 1, 2, . . . be random variables taking values in the set X. (N) Triplets ξ¯ε,i,n = ξε,i,n (t), t ≥ 0 , κε,i,n, ηε,i,n , i ∈ X, n = 1, 2, . . . and the random variable ηε are mutually independent. (O) Joint distributions of random variables ξε,i,n (tk ), k = 1, . . . , r and κε,i,n, ηε,i,n do not depend on n ≥ 1, for every i ∈ X and tk ∈ [0, ∞), k = 1, . . . , r, r ≥ 1. The above assumption of measurability for processes ξ¯ε,i,n is absolutely similar to those formulated for processes ξ¯ε,n in Sect. 2.1.1.1. Let us define recurrently stochastic sequences of switching random indices, ηε,n = ηε,ηε, n−1,n, n = 1, 2, . . . , ηε,0 = ηε,

(10.1)

ζε,n = κε,ηε,0,1 + · · · + κε,ηε, n−1,n, n = 1, 2, . . . , ζε,0 = 0,

(10.2)

regeneration times,

and modulated multi-alternating regenerative process (ξε (t), ηε (t)), t ≥ 0 with two components, ξε (t) = ξε,ηε, n−1,n (t − ζε,n−1 ) and ηε (t) = ηε,n−1, for t ∈ [ζε,n−1, ζε,n ), n = 1, 2, . . . .

(10.3)

From the above model assumptions (K)–(O) it follows that the random sequence ηε,n, n = 0, 1, . . . is a homogeneous Markov chain with the phase space X, initial distribution p¯ε = pε,i = P{ηε,0 = i}, i ∈ X , and transition probabilities, pε,i j = P{ηε,1 = j/ηε,0 = i} = P{ηε,i,1 = j}, i, j ∈ X.

(10.4)

The above model assumptions (K)–(O) also imply that the random sequence (ηε,n, κε,n ), n = 0, 1, . . . (here, κε,0 = 0) is a Markov renewal process, with the phase space X × [0, ∞), initial distribution q¯ε = qε,i = P{ηε,0 = i, κε,0 = 0}, i ∈ X , and transition probabilities,

10.1 MARP

189

Q ε,i j (t) = P{ηε,1 = j, κε,1 /ηε,0 = i} = P{ηε,i,1 = j, κε,i,1 ≤ t}, t ≥ 0, i, j ∈ X.

(10.5)

The above model assumptions (K)–(O) also imply that the stochastic process ηε (t), t ≥ 0 is, for every ε ∈ (0, 1], a semi-Markov process with the phase space X and transition probabilities Q ε,i j (t), t ≥ 0, i, j ∈ X. The process ηε (t) plays the role of a modulating semi-Markov process for the regenerative component ξε (t). Let us also introduce the distribution function of the sojourn time in state i ∈ X for the semi-Markov process ηε (t), Fε,i (t) = P{κε,1 ≤ t/ηε,0 = i}

Q ε,i j (t), t ≥ 0. = P{κε,1,i ≤ t} =

(10.6)

j ∈X

Ergodic theorems for perturbed multi-alternating regenerative processes are based on limit theorems for hitting times for perturbed semi-Markov processes, presented in Chaps. 81 –121 , and ergodic theorems for perturbed regenerative and alternating regenerative processes, presented in Chaps. 2–9. We prefer to formulate conditions imposed on perturbed multi-alternating regenerative processes as analogues of the conditions G–K used in Chaps. 81 –121 , specifying similar conditions used in Chaps. 2–9. The condition G takes in this case the following form: G1 : (a) pε,i j > 0, ε ∈ (0, 1] or pε,i j = 0, ε ∈ (0, 1], for every i, j ∈ X, (b) Fε,i (0) = 0, for i ∈ X, ε ∈ (0, 1]. The condition G1 (b) guarantees that Pi {limn→∞ ζε,n = ∞} = 1, for any i ∈ X and ε ∈ (0, 1]. Therefore, the multi-alternating regenerative process (ξε (t), ηε (t)) is well defined on the time interval [0, ∞), for every ε ∈ (0, 1]. The condition H takes in this case the following form: H1 : For any i, j ∈ X, there exist ni j ≥ 1 and a chain of states i = r0, r1, . . . , rni j = j  such that 1≤l ≤ni j p1,rl−1 rl > 0. The conditions G1 (a) and H1 imply that the phase space X of the embedded Markov chain ηε,n is one class of communicative states, for every ε ∈ (0, 1]. We introduce sets, for i ∈ X and ε ∈ [0, 1], Yε,i = { j ∈ X : pε,i j > 0}.

(10.7)

It is useful to note that the condition G2 (a) implies that, for i ∈ X and ε ∈ (0, 1], Yε,i = Y1,i .

(10.8)

Let us consider Z∞ as the space of Borel measurable functions z(·) = z(t), t ≥ 0

defined on the interval [0, ∞) and taking values in the space Z (sets {t ∈ [0, ∞) : z(t) ∈ A} ∈ B+ for A ∈ BZ ) and BZ∞ as the σ-algebra of subsets of the space Z∞ (the

190

10 Perturbed MARP

minimal σ-algebra containing all cylindric sets {z(·) ∈ Z∞ : z(t1 ) ∈ A1, . . . , z(tn ) ∈ An }, for t1, . . . , tn ∈ [0, ∞), A1, . . . , An ∈ BZ, n ≥ 1. Consider the following joint distributions, for i ∈ X and ε ∈ (0, 1]: Q ε,i j (t, C) = P{ ξ¯ε,i,1 ∈ C, κε,i,1 ≤ t, ηε,i,1 = j}, C ∈ BZ∞ , t ≥ 0, j ∈ X,

(10.9)

and Fε,i (t, C) = P{ ξ¯ε,i,1 ∈ C, κε,i,1 ≤ t} =



Q ε,i j (t, C), C ∈ BZ∞ , t ≥ 0.

(10.10)

j ∈X

Also, let us consider the conditional joint distribution defined for i, j ∈ X and ε ∈ (0, 1], Fε,i j (t, C) = P{ ξ¯ε,i,1 ∈ C, κε,i,1 ≤ t/ηε,i,1 = j}, C ∈ BZ∞ , t ≥ 0.

(10.11)

If j ∈ Y1,i, i ∈ X, then, for ε ∈ (0, 1], Fε,i j (t, C) = p−1 ε,i j Q ε,i j (t, C), t ≥ 0, C ∈ BZ∞ , t ≥ 0.

(10.12)

¯ 1,i, i ∈ X, an arbitrary distribution Fε,i j (t, C) can play the role of Fε,i j (t, C). If j ∈ Y We use the standard variant and choose in this case, for ε ∈ (0, 1], Fε,i j (t, C) = Fε,i (t, C), C ∈ BZ∞ , t ≥ 0

(10.13)

Fε,i j (t) = Fε,i j (t, Z∞ ) = Fε,i (t), t ≥ 0.

(10.14)

and, thus, ¯ 1,i, i ∈ X has no effect on the Note that the choice of Fε,i j (t, C) for j ∈ Y distribution Q ε,i j (t, C) and the transition probabilities Q ε,i j (t) = Q ε,i j (t, Z∞ ). In any case, the following relation takes place for C ∈ BZ∞ , t ≥ 0 and ε ∈ (0, 1]:  Fε,i j (t, C)pε,i j for j ∈ Y1,i, i ∈ X, (10.15) Q ε,i j (t, C) = ¯ 1,i, i ∈ X, 0 for j ∈ Y and, thus, for t ≥ 0 and ε ∈ (0, 1],  Fε,i j (t)pε,i j for j ∈ Y1,i, i ∈ X, Q ε,i j (t) = ¯ 1,i, i ∈ X. 0 for j ∈ Y

(10.16)

¯ 1,i, i ∈ X It follows from the above remarks that the choice of Fε,i j (t, C) for j ∈ Y also does not affect the finite-dimensional distributions of the multi-alternating regenerative process (ξε (t), ηε (t)), t ≥ 0, for ε ∈ (0, 1]. In what follows, we always use the choice indicated in the relation (10.13). 10.1.1.2 Multi-Alternating Regenerative Processes with Transition Periods. Let us assume that the model assumption (O) holds only for n ≥ 2.

10.1 MARP

191

In this case, the process (ξε (t), ηε (t)) defined using the recurrent relations (10.1)– (10.3) can be called multi-alternating regenerative processes with a transition period [0, ζε,1 ). The shifted process (ξε(1) (t), ηε(1) (t)) = (ξε (ζε,1 + t), ηε (ζε,1 + t)) ≥ 0 is a usual alternating regenerative process. Moreover, it is possible to generalise the model and assume that the random  = {1, . . . , m}  that differs from the phase space X, variable ηε has a phase space X  is used for the and the family of triplets ξ¯ε,1,i = ξε,1,i (t), t ≥ 0 , κε,1,i, ηε,1,i , i ∈ X  are measurable processes with transition period. Here, ξ¯ε,1,i = ξε,1,i (t), t ≥ 0 , i ∈ X  are non-negative random variables, and ηε,1,i, i ∈ X  the phase space Z, κε,1,i, i ∈ X are random variables taking values in the phase space X. As above, it is assumed that the stochastic triplets ξ¯ε,1,i = ξε,1,i (t), t ≥  and ξ¯ε,n,i = ξε,n,i (t), t ≥ 0 , κε,n,i, ηε,n,i , n ≥ 2, i ∈ X, 0 , κε,1,i, ηε,1,i , i ∈ X and the random variable ηε are mutually independents and the stochastic triplets ξ¯ε,n,i = ξε,n,i (t), t ≥ 0 , κε,n,i, ηε,n,i , n ≥ 2 are probabilistic copies of the triplet ξ¯ε,2,i = ξε,2,i (t), t ≥ 0 , κε,2,i, ηε,2,i , for every i ∈ X (in the sense of fulfilling the model assumption (O)). The corresponding multi-alternating regenerative process (ξε (t), ηε (t)) with transition period [0, ζε,1 ) is defined using the same recurrent relations (10.1)–(10.3). A model of multi-alternating regenerative process with transition period and  and X arises naturally when the algorithm of phase space two phase spaces X reduction of a modulating semi-Markov process is applied to a standard (without a transition period) multi-alternating regenerative process. In this case, the original multi-alternating regenerative process has the modulating semi-Markov process with phase space X, while the resulting multi-alternating regenerative process has the modulating semi-Markov process with a reduced phase space k¯h X with m − h states. In this case the random variable ζε,1 is the first time of hitting in the reduced phase  and X is played by the spaces X and k¯ X. space k¯h X. The role of the phase spaces X h In the case where h = m − 2, the reduced phase space k¯ m−2 X is a two-state set and the corresponding reduced multi-alternating regenerative process is an alternating regenerative process. In the case where h = m − 1, the reduced phase space k¯ m−1 X is a one-state set and the corresponding multi-alternating regenerative process is a regenerative process.

10.1.2 Perturbation Conditions for Multi-Alternating Regenerative Processes 10.1.2.1 Perturbation Conditions for Standard Multi-Alternating Regenerative Processes. In what follows we assume that the conditions G1 and H1 hold. The condition I takes in this case the following form: I1 : pε,i j → p0,i j as ε → 0, for i, j ∈ X.

192

10 Perturbed MARP

Since the matrix Pε = pε,i j  is stochastic, it follows from the conditions H1 and I1 that the matrix P0 = p0,i j  is also stochastic. Let η0,n, n = 0, 1, . . . be a Markov chain with the phase space X and the matrix of transition probabilities P0 . The condition I allows us to interpret the Markov chain ηε,n , for ε ∈ (0, 1], as a perturbed version of the Markov chain η0,n . The case, where the phase space X is one class of communicative states plus possibly a class of transient states for the Markov chain η0,n , refers to the model with regular perturbations. The case, where the phase space X is divided into several closed classes of communicative states plus possibly a class of transient states for the Markov chain η0,n , refers to the model with singular perturbations. The condition IH takes the following form: IH,1 : The functions p ·,i j , j ∈ Y1,i, i ∈ X belong to a complete family of asymptotically comparable functions H. The notion of complete families of asymptotically comparable function is introduced in Chap. 81 , where one can find examples of such families and operational rules for asymptotically comparable functions. Short comments concerned complete families of asymptotically comparable functions are also given in Sect. B.2.1. Note that the condition IH,1 implies the fulfilment of the condition I1 . According to the relations (10.12) and (10.14), for t ≥ 0 and ε ∈ (0, 1],  Q ε,i j (t)/pε,i j for t ≥ 0, j ∈ Y1,i, i ∈ X, (10.17) Fε,i j (t) = ¯ 1,i, i ∈ X. for t ≥ 0, j ∈ Y Fε,i (t) We assume that the following variant of condition J holds: J1 : (a) Fε,i j (·uε,i ) ⇒ F0,i j (·) as ε → 0, for j ∈ Y1,i, i ∈ X, (b) F0,i j (·) is a nonarithmetic distribution function without singular component, for j ∈ Y1,i, i ∈ X, (c) uε,i ∈ (0, ∞), ε ∈ (0, 1], for i ∈ X. The condition J1 is a particular case of the condition J. Indeed, the assumption that the limiting distribution functions F0,i j (·), j ∈ Y1,i, i ∈ X are non-arithmetic and have not singular component is used in the condition J1 . The weaker assumption, F0,i j (0) < 1, j ∈ Y1,i, i ∈ X, is used in the condition J. The condition J1 can be formulated in equivalent form in terms of the Laplace transforms of the distribution functions Fε,i j (·), for i, j ∈ X and ε ∈ (0, 1], ∫ ∞ φε,i j (s) = e−su Fε,i j (du), s ≥ 0. (10.18) 0

¯ 1,i, i ∈ X and ε ∈ (0, 1], According to the relation (10.17), for j ∈ Y ∫ ∞ φε,i j (s) = φε,i (s) = e−su Fε,i (du), s ≥ 0. 0

The following condition is equivalent to the condition J1 :

(10.19)

10.1 MARP

193

J◦1 : (a) φε,i j (s/uε,i ) → φ0,i j (s) as ε → 0, for s ≥ 0 and j ∈ Y1,i, i ∈ X, (b) ∫∞ φ0,i j (·) = 0 e−su F0,i j (du), s ≥ 0 is the Laplace transform of non-arithmetic distributions function without singular component F0,i j (·), for j ∈ Y1,i, i ∈ X, (c) uε,i ∈ (0, ∞), ε ∈ (0, 1], for i ∈ X. The conditions G1 and I1 imply that, for i ∈ X, Y0,i = { j ∈ X : p0,i j > 0} ⊆ Y1,i .

(10.20)

The conditions G1 , I1 , J1 , and the relation (10.20) also imply that, for i ∈ X,



Fε,i (·uε,i ) = Q ε,i j (·uε,i ) = Fε,i j (·uε,i )pε,i j j ∈Y1, i



⇒

j ∈Y1, i

F0,i j (·)p0,i j =

j ∈Y1, i



F0,i j (·)p0,i j

j ∈Y0, i

= F0,i (·) as ε → 0

(10.21)

or, equivalently,

φε,i (suε,i ) =

φε,i j (suε,i )pε,i j

j ∈Y1, i

→



φ0,i j (s)p0,i j =

j ∈Y1, i

∫

= φ0,i (s) =



φ0,i j (s)p0,i j

j ∈Y0, i ∞

e−su F0,i (du) as ε → 0, for s ≥ 0.

(10.22)

0

Let us define, for i, j ∈ X and t ≥ 0,  F0,i j (t)p0,i j for j ∈ Y0,i, i ∈ X, Q0,i j (t) = ¯ 0,i, i ∈ X. 0 for j ∈ Y

(10.23)

Obviously, the functions Q0,i j (t), t ≥ 0, i, j ∈ X can serve as semi-Markov transition probabilities, i.e., these functions are non-negative, continuous from the right, and Q0,i j (t) → Q0,i j (∞) = p0,i j as t → ∞, for i, j ∈ X. The following useful lemma takes place. Lemma 10.1 The conditions G1 , I1 , and J1 imply that, for i, j ∈ X, Q ε,i j (·uε,i ) ⇒ Q0,i j (·) as ε → 0,

(10.24)

Q ε,i j (∞) = pε,i j → p0,i j = Q0,i j (∞) as ε → 0.

(10.25)

and Proof The relation (10.25) follows from the relation (10.23) and the condition I1 . The conditions I1 and J1 and the relation, Y0,i ⊆ Y1,i , imply that, for j ∈ Y0,i , i ∈ X,

194

10 Perturbed MARP

Q ε,i j (·uε,i ) = Fε,i j (·uε,i )pε,i j ⇒ F0,i j (·)p0,i j = Q0,i j (·) as ε → 0.

(10.26)

¯ 0,i, i ∈ X and t ≥ 0, Also, the condition G1 and I1 imply that, for j ∈ Y Q ε,i j (t) ≤ Q ε,i j (∞) = pε,i j → p0,i j = 0 as ε → 0,

(10.27)

Q ε,i j (·) ⇒ Q0,i j (·) = 0(·) ≡ 0 as ε → 0.

(10.28)

and, thus, 

The proof is complete.

Let us also introduce the expectations of inter-jump times, for i, j ∈ X and ε ∈ (0, 1], ∫ ∞ tFε,i j (dt) for j ∈ Y1,i, i ∈ X, (10.29) fε,i j = 0 ¯ 1,i, j ∈ X, fε,i for j ∈ Y where, for i ∈ X,

∫ fε,i =

∞

0



tFε,i (dt) =

fε,i j pε,i j .

(10.30)

j ∈Y1, i

We assume that the following variant of the condition K holds: K1 : (a) ∫fε,i j < ∞, j ∈ Y1,i, i ∈ X, for every ε ∈ (0, 1], (b) fε,i j /uε,i → f0,i j ∞ = 0 tF0,i j (dt) < ∞ as ε → 0, for j ∈ Y1,i, i ∈ X. The relation (10.30) and the conditions I1 , K1 imply that, for i ∈ X and ε ∈ (0, 1], fε,i < ∞,

(10.31)

and

fε,i j fε,i = pε,i j uε,i j ∈Y uε,i 1, i



→ f0,i j p0,i j = f0,i j p0,i j j ∈Y1, i

∫

= f0,i =

j ∈Y0, i ∞

tF0,i (dt) < ∞ as ε → 0.

(10.32)

0

Let us introduce the expectations, for i, j ∈ X, and ε ∈ (0, 1], eε,i j = Ei ζε,1 I(ηε,1 = j) = E κε,i,1 I(ηε,i,1 = j) =

∫ 0

∞

sQ ε,i j (ds).

(10.33)

10.1 MARP

195

It follows from the condition K1 (a) and the relation (10.33) that, for i, j ∈ X, and ε ∈ (0, 1], ∫ eε,i j =

∞

0

sFε,i j (ds)pε,i j = fε,i j pε,i j < ∞.

(10.34)

Let us also introduce the expectations, for i ∈ X and ε ∈ (0, 1],

eε,i = Ei ζε,1 = eε,i j =



j ∈X

fε,i j pε,i j =

j ∈X



fε,i j pε,i j .

(10.35)

j ∈Y1, i

The relation (10.34) implies that, for i ∈ X and ε ∈ (0, 1], eε,i < ∞.

(10.36)

Let us define, for i, j ∈ X,  e0,i j =

f0,i j p0,i j for j ∈ Y0,i, i ∈ X, ¯ 0,i, i ∈ X. 0 for j ∈ Y

(10.37)

The following useful lemma takes place. Lemma 10.2 The conditions G1 , I1 , J1 , and K1 imply that, for i, j ∈ X, eε,i j /uε,i → e0,i j as ε → 0.

(10.38)

Proof The conditions I1 , K1 , and the relations (10.37) and Y0,i ⊆ Y1,i, i ∈ X imply that the following relation holds, for j ∈ Y1,i, i ∈ X: eε,i j fε,i j = pε,i j → f0,i j p0,i j = e0,i j as ε → 0. uε,i uε,i

(10.39)

The condition G1 implies that Q ε,i j (∞) = pε,i j = 0, ε ∈ (0, 1], and, thus, eε,i j = ¯ 1,i, i ∈ X. 0, ε ∈ (0, 1], for j ∈ Y ¯ 0,i , for i ∈ X, the relation (10.37) implies that e0,i j = 0, for ¯ Also, the set Y1,i ⊆ Y ¯ j ∈ Y1,i, i ∈ X. ¯ 1,i, i ∈ X, Therefore, for j ∈ Y eε,i j /uε,i = 0 → e0,i j = 0 as ε → 0. The proof is complete. Finally, the conditions L and LH take the following forms: L1 : uε,i → u0,i ∈ (0, ∞] as ε → 0, for i ∈ X and

(10.40) 

196

10 Perturbed MARP

LH,1 : The functions u ·,i, i ∈ X belong to the complete family of asymptotically comparable functions H. The object of our interest is joint distributions, Pε,i j (t, A) = Pi {ξε (t) ∈ A, ηε (t) = j}, A ∈ BZ, i, j ∈ X, t ≥ 0.

(10.41)

The probabilities Pε,i j (t, A), i ∈ X are, for every A ∈ BZ, j ∈ X, measurable functions of t ≥ 0, which are the only bounded solution to the following system of renewal type equations: Pε,i j (t, A) = I(i = j)qε,i (t, A)

∫ t Pε,k j (t − s, A)Q ε,ik (ds), t ≥ 0, i ∈ X + k ∈X

(10.42)

0

where, for A ∈ BZ, t ≥ 0, i ∈ X, qε,i (t, A) = Pi {ξε (t) ∈ A, ηε (t) = i, ζε,1 > t} = P{ξε,i,1 (t) ∈ A, κε,i,1 > t}.

(10.43)

It is also worth noting that the asymptotic analysis of the probabilities Pε,i j (t, A) =

Pi {ξε (t) ∈ A, ηε (t) = j} can be reduced to an asymptotic analysis of the formally simpler probabilities, Pi {ξε (t) ∈ A}. Indeed, one can always consider the extended

multi-alternating regenerative process (ξε (t), ηε (t)) with the extended regenerative component ξε (t) = (ξε (t), ηε (t)). The process ξε (t) has the phase space Z = Z × X, with the σ-algebra of measurable subsets BZ = {C = ∪i ∈B (Ai × {i}) : Ai ∈ BZ, i ∈ B ⊆ X}. In this case, for A ∈ BZ, t ≥ 0, i, j ∈ X, Pε,i j (t, A) = Pi {ξε (t) ∈ A, ηε (t) = j} = Pi { ξε (t) ∈ A × { j}}.

(10.44)

Taking into account the above remarks, we will further consider the probabilities, Pε,i (t, A) = Pi {ξε (t) ∈ A}, A ∈ BZ, t ≥ 0, i ∈ X.

(10.45)

The probabilities Pε,i (t, A), i ∈ X are, for every A ∈ BZ , measurable functions of t ≥ 0, which are the only bounded solution to the following system of renewal type equations (an analogue of the system of renewal type equations (10.42)): Pε,i (t, A) = qε,i (t, A)

∫ t Pε,k (t − s, A)Q ε,ik (ds), t ≥ 0, i ∈ X, + k ∈X

0

where, for A ∈ BZ, t ≥ 0, i ∈ X,

(10.46)

10.1 MARP

197

qε,i (t, A) = Pi {ξε (t) ∈ A, ζε,1 > t} = P{ξε,i,1 (t) ∈ A, κε,i,1 > t}.

(10.47)

Note that the function qε,i (t, A) is the same for both systems of renewal type equations (10.42) and (10.46). According to the relation (10.47), the functions qε,i (tuε,i, A), t ∈ R+, A ∈ BZ belong to the class P[BZ ], for i ∈ X, ε ∈ (0, 1]. Moreover, the function qε,i (tuε,i, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − Fε,i (tuε,i ), t ≥ 0, for i ∈ X and ε ∈ (0, 1], i.e., qε,i (tuε,i, Z) = 1 − Fε,i (tuε,i ), for t ∈ R+ .

(10.48)

We also assume that the following perturbation condition holds: R: There exist functions q0,i (t, A), t ≥ 0, A ∈ BZ , i ∈ X, which belong to the class P[BZ ], a class of sets Γ ⊆ BZ , and Borel sets U[q ·,i (·u ·,i, A)], A ∈ Γ, i ∈ X such that: (a) the function q0,i (t, A), t ∈ R+, A ∈ BZ is consistent with the tail probabilus ity function 1−F0,i (t), t ≥ 0, for i ∈ X; (b) the functions qε,i (·uε,i, A) −→ q0,i (·, A) as ε → 0, for points s ∈ U[q ·,i (·u ·,i, A)], A ∈ Γ, i ∈ X; (c) m(U¯ [q ·,i (·u ·,i, A)]) = 0, for A ∈ Γ, i ∈ X; (d) the function q0,i (·, A) is continuous almost everywhere with respect to the Lebesgue measure m(·) on B+ , for A ∈ Γ, i ∈ X. It should be noted that the notation U[q ·,i (·u ·,i, A)] is used to show that this set of convergence is, actually, determined by the family of functions qε,i (·uε,i, A), ε ∈ (0, 1]. It is useful to make a few comments about the above perturbation condition. In the light of the relations (10.48), the consistency condition R (a) is natural. The consistency relation (10.48) implies that, for every ε ∈ (0, 1] and A ∈ BZ, i ∈ X, the function qε,i (·uε,i, A) is majorised by the tail probability function 1−Fε,i (·uε,i ) on interval [0, ∞), i.e., qε,i (tuε,i, A) ≤ 1 − Fε,i (tuε,i ), for t ∈ R+ .

(10.49)

The condition R (a) implies that the following majorisation relation similar to (3.39) takes place, for A ∈ BZ, i ∈ X: q0,i (t, A) ≤ 1 − F0,i (t), for t ∈ R+ .

(10.50)

Remark 10.1 As mentioned in Sect. 10.1.1.1, the choice of the distribution functions ¯ i,1, i ∈ X does not affect the transition probabilities Q ε,i j (t), t ≥ Fε,i j (·), j ∈ Y 0, i, j ∈ X of the modulating semi-Markov process ηε (t) and the finite-dimensional distributions of the multi-alternating regenerative process (ξε (t), ηε (t)). It is natural to expect that the conditions G1 –K1 and R should also not depend on the choice ¯ i,1, i ∈ X. This is indeed so, since all of the distribution functions Fε,i j (·), j ∈ Y quantities appearing in the above conditions do not depend on these distribution functions.

198

10 Perturbed MARP

10.1.2.2 Structure of the Class Γ. The condition R implies that the class Γ can always be extended to its maximal form such that Γ would be closed with respect to the operations of the union of disjoint sets, the difference of sets connected by the relation of inclusion, and the complement. Let Γ ⊆ BZ be the maximal class of sets A ∈ BZ , for which condition R holds. Lemma 10.3 Let the conditions I1 , J1 , and R be satisfied. Then, the maximal class Γ: (a) contains the phase space Z and (b) is closed with respect to the operations of union for disjoint sets, difference for sets connected by the relation of inclusion, and complement. The proof of this lemma is similar to the proof of Lemma 3.4. If the space Z = Z × X, then the corresponding class Γ includes all sets of the form A = ∪i ∈B Ai × {i}, where Ai ∈ Γ, i ∈ B ⊆ X. 10.1.2.3 Ergodic Theorems for Alternating Regenerative Processes with Degenerated Switching Random Variables. Let us consider the case, where the switching random variables ηε,i,n = i, n ≥ 1 with probability 1, for some i ∈ X and each ε ∈ (0, 1]. Let us also consider, for every ε ∈ (0, 1], i ∈ X, the standard regenerative process, ξε,i (t) = ξε,i,n (t − ζε,i,n−1 ), for t ∈ [ζε,i,n−1, ζε,in ), n = 1, 2, . . .

(10.51)

with the regeneration times, ζε,i,n = κε,i,1 + · · · + κε,i,n, n = 1, 2, . . . , ζε,i,0 = 0.

(10.52)

The condition G1 , implies that, for ε ∈ (0, 1], i = 1, 2, the random variables P

ζε,i,n −→ ∞ as ε → 0 and, thus, the process ξε,i (t) is well defined on the time interval [0, ∞). The probabilities pε,i (t, A) = P{ξε,i (t) ∈ A} ∈ [0, 1], t ≥ 0 are, for ε ∈ (0, 1], A ∈ BZ, i ∈ X, the unique in class L solution of the renewal equation, ∫ t pε,i (t, A) = qε,i (t, A) + pε,i (t − s, A)Fε,i (ds), t ≥ 0. (10.53) 0

According to the condition R, the function q0,i (t, A) belongs to class P[Z], for every i ∈ X. According to the condition J1 , F0,i (·), i ∈ X are proper distribution functions, which is not concentrated at zero. Thus, the following standard renewal equation can be written, for A ∈ Γ and i ∈ X: ∫ t p0,i (t, A) = q0,i (t, A) + p0,i (t − s, A)F0,i (ds), t ≥ 0. (10.54) 0

It has the unique solution p0,i (t, A), t ≥ 0 that belongs to class L. Moreover, p0,i (t, A) ∈ [0, 1], t ≥ 0. The conditions I1 , J1 , K1 , and R imply that the distribution functions Fε,i (s), their first moments eε,i , and the functions qε,i (t) converge in some sense to the

10.1 MARP

199

corresponding limiting distribution function F0,i (s), its first moment e0,i , and the function q0,i (t) as ε → 0. This allows us to consider the equation (10.53) for ε ∈ (0, 1] as a perturbed version of the equation (10.54). The conditions G1 –K1 , and R imply that the functions q0,i (·, A), A ∈ Γ, i ∈ X, belong to the class L, are directly Riemann integrable on [0, ∞), and the first moments e0,i < ∞, i ∈ X. Therefore, for every A ∈ Γ, i ∈ X, the conditions of Theorem A.2 are satisfied for the renewal equations (10.53). According to this theorem, the following asymptotic relation takes place for every A ∈ Γ, i ∈ X and any 0 ≤ tε → ∞ as ε → 0: ∫ ∞ 1 pε,i (tε, A) → π0,i (A) = q0,i (s, A)ds as t → ∞. (10.55) e0,i 0 The stationary distributions π0,i (·), i = 1, 2 are used in formulas for stationary distributions in the ergodic theorems for perturbed multi-alternating regenerative processes presented in Chap. 13. Let, for A ∈ BZ and i ∈ X, ∫ ∞ 1 q0,i (s, A)m(ds). (10.56) π0,i (A) = e0,i 0 The finite Lebesgue integral on right hand side of relation (10.56) exists since, by the conditions I1 , J1 , K1 , and R, the functions q0,i (·, A), i ∈ X belong to the class L and are ∫majorised, for every A ∈ BZ , by the tail probability function 1 − F0,i (·), ∞ for which 0 (1 − F0,i (s))m(ds) = e0,i < ∞, i ∈ X. Moreover, π0,i (A), A ∈ BZ is, for i ∈ X, a probability measure on the σ-algebra BZ , since according to the condition R, the function q0,i (s, A), s ∈ R+, A ∈ BZ belongs to class P[Z], for i ∈ X and is consistent with the tail probability function 1 − F0,i (t), t ∈ R+ . Note that the Lebesgue integration is used in expression on the right hand side of the relation (10.56) instead of the direct Riemann integration used in the expression on the right hand side of the relation (10.55). According to the condition R, the functions q0,i (·, A) are directly Riemann integrable for A ∈ Γ, and, thus, the Lebesgue integration in (10.56) can be replaced by the direct Riemann integration, for A ∈ Γ, as is done in the relation (10.55), that is, for A ∈ Γ, i ∈ X, ∫ ∞ ∫ ∞ 1 1 q0,i (s, A)m(ds) = q0,i (s, A)ds. (10.57) π0,i (A) = e0,i 0 e0,i 0 This measure is the limiting stationary distribution for the perturbed regenerative processes ξε,i (t), for i ∈ X, for the case where ε → 0. From the above remarks it follows that the limits π0,i (A), A ∈ Γ, defined by the relation (2.47) for the maximal class Γ (see, Sect. 10.1.2.2) have the following properties, for i ∈ X: (a) π0,i (Z) = 1, (b) π0,i (A ∪ A) = π0,i (A) + π0,i (A), for

200

10 Perturbed MARP

A, A ∈ Γ.A ∩ A = ∅,, (c) π0,i (A \ A) = π0,i (A) − π0,i (A), for A, A ∈ Γ, A ⊆ ¯ = 1 − π0,i (A), for A ∈ Γ. A = ∅, (d) π0,i ( A) Moreover, if the class Γ is a σ-algebra, then the function π0,i (A), A ∈ Γ defined by relation (10.55) is a probability measure on Γ, for i ∈ X. 10.1.2.5 Perturbation Conditions for Multi-Alternating Regenerative Processes with Transition Period. In this case, all quantities appearing in the conditions G1 , H1 I1 , IH,1 , J1 (J◦1 ), K1 , and R, the systems of renewal type equations (10.42), and the quantities defined in the relations (10.43) and (10.47) should be defined using the shifted sequence of triplets ξ¯ε,i,2 = ξε,i,2 (t), t ≥ 0 , κε,i,2, ηε,i,2 , i ∈ X. It is also natural to index the above mentioned quantities by the upper index (1) , for example, (1) to use the notations Pε,i (t, A) = Pi {ξε (t) ∈ A} and Pε,i (t, A) = Pi {ξε(1) (t) ∈ A}, etc. Also, some condition related to transition period and similar to the condition P 2 should be assumed to hold for transition probabilities pε,i j = P{ηε,i,1 = j}, i, j ∈ X and Q ε,i j (t) = P{κε,i,1 ≤ t, ηε,i,1 = j}, t ≥ 0, i, j ∈ X. In the generalised case (with  and X) considered in Sect. 10.1.1.2, this condition takes the two phase spaces X following form:  (a) pε,i j → p0,i j as ε → 0, for i ∈ X,  j ∈ X, (b) Q ε,i j (·uε,i ) ⇒ Q 0,i j (·) as P:  j ∈ X, where Q 0,i j (·) is a proper or improper distribution ε → 0, for i ∈ X,  j ∈ X, (c) function on interval [0, ∞) such that Q 0,i j (∞) = p0,i j , for i ∈ X, uε,i ∈ (0, ∞), ε ∈ (0, 1], for i ∈ X. Note that the case, where some or even all distribution functions Q 0,i j (·) = I(u ≥ 0) p0,i j is also admissible in condition P 3 (b). For shifted multi-alternating regenerative processes ξε(1) (t) (which have no transition periods), the ergodic theorems presented in Chap. 13 gave asymptotic relations (1) (tε uε, A), for 0 ≤ tε → ∞ as ε → 0 and some time for the probabilities Pε,i compression factors uε such that 0 < uε → u0 ∈ (0, ∞] as ε → 0. For the case, where ξε (t) are multi-alternating regenerative processes with transition periods, the additional condition of asymptotic comparability for the time compression factors uε,i, i ∈ X and uε should be also assumed to hold:  uε,i /uε → w i ∈ [0, ∞) as ε → 0, for i ∈ X.  X:  and P(1) (t, A), t ≥ 0, i ∈ X, In this case, the probabilities Pε,i (t, A), t ≥ 0, i ∈ X ε,i for A ∈ BZ , are connected by the following renewal type relation: Pε,i (tuε, A) = P{ξε (tuε ) ∈ A, ζε,1 > tε uε }

∫ t (1)   Pε, + j ((t − s)uε , A)Q ε,i j (uε ds), t ≥ 0, i ∈ X. j ∈X

(10.58)

0

 make it possible to The relation (10.58) and the perturbation conditions P and X obtain ergodic relations for the probabilities Pε,i (tε uε, A) using the corresponding (1) ergodic relations for the probabilities Pε, j (tε uε , A).

10.2 MARP with removed virtual transitions

201

10.2 Multi-Alternating Regenerative Processes with Removed of Virtual Transitions In this section, asymptotic procedures of aggregation of regeneration times, based on total and partial removing of virtual transitions for modulating semi-Markov processes, are described.

10.2.1 Procedure of Total Removing of Virtual Transitions for Modulating Semi-Markov Processes 10.2.1.1 Multi-Alternating Regenerative Processes with Totally Removed Virtual Transitions for Modulating Semi-Markov Processes. Let us assume that ε ∈ (0, 1] and the conditions G1 and H1 hold. Let us define for r = 0, 1, . . . the stopping time, which is the first after r moment of change of state ηε,r by the Markov chain ηε,n , θ˜ε [r] = min(n > r : ηε,n  ηε,r ).

(10.59)

We also define stopping times that are successive moments of state change by the Markov chain ηε,n , ρε,n = θ ε [ρε,n−1 ], n = 1, 2, . . . , where ρε,0 = 0.

(10.60)

Let us now construct a new Markov renewal process (η˜ε,n, κ˜ε,n ), n = 0, 1, . . ., with the phase space X × [0, ∞),  (ηε,0, 0) for n = 0, ρε, n (η˜ε,n, κ˜ε,n ) = (η (10.61) ε,ρε, n , l=ρε, n−1 +1 κε,l ) for n = 1, 2, . . . . We also can define the corresponding modulating semi-Markov process, η˜ε (t) = ηε, ν˜ ε (t), t ≥ 0,

(10.62)

ζ˜ε,n = κ˜ε,1 + · · · + κ˜ε,n, n = 1, 2, . . . , ζ˜ε,0 = 0,

(10.63)

where, are the corresponding moments of jumps, and, ν˜ε (t) = max(n ≥ 1 : ζ˜ε,n ≤ t),

(10.64)

is the number of jumps in the interval [0, t] for the above semi-Markov process. Let us consider a two component stochastic process, (ξ˜ε (t), η˜ε (t)) = (ξε (t), η˜ε (t)), t ≥ 0.

(10.65)

202

10 Perturbed MARP

This is a multi-alternating regenerative process. It has a regenerative component the same as the original multi-alternating regenerative process (ξε (t), ηε (t)), but has a new modulating semi-Markov process η˜ε (t). Moreover, it follows from the definition of the semi-Markov process η˜ε (t) that it has trajectories a.s. (almost sure) identical to the trajectories of the process ηε (t), i.e., η˜ε (t) = ηε (t), t ≥ 0 and, therefore, the new and original multivariate regenerative processes also have a.s. identical trajectories, that is, for i ∈ X, Pi {(ξ˜ε (t), η˜ε (t)) = (ξε (t), ηε (t)), t ≥ 0} = 1.

(10.66)

The difference between these processes lies in their regeneration times ζ˜ε,n, n = 0, 1, . . . and ζε,n, n = 0, 1, . . .. It is worth noting here that, in fact, most of the identity relations that define random variables and processes, for example, the relation (10.62) only a.s. take place. The point is that the relation (10.59) determines finite random variables and, therefore, trajectories of the process ηε (t), t ≥ 0 are well defined only a.s. In the book, we mostly write out equalities that define random variables and processes in the usual way (omitting a.s. appendix), bearing in mind that this “almost certainly” aspect in the defining equalities does not affect the distributions of these random variables and processes, as well as the algorithms for their calculation. Let us also describe a procedure of constructing a family of stochastic triplets satisfying the model assumptions (K)–(O) and such that a multi-alternating regenerative process built using these triplets and the relations (10.1)–(10.3) would have the same finite-dimensional distributions as the multi-alternating regenerative process (ξ˜ε (t), η˜ε (t)).   (t), t ≥ 0 , κ  , η  stochastic triplet ξ¯ = = ξε,i Let us denote by ξ¯ε,i ε ε,i ε,i ξ˜ε (t), t ≥ 0 , κ˜ε,1, η˜ε,1 , for the case, where the initial state ηε = i, for i ∈ X.     = ξε,i,n (t), t ≥ 0 , κε,i,n , ηε,i,n

, Let also a family of stochastic triplets ξ¯ε,i,n  i ∈ X, n = 1, 2, . . . and a random variable ηε satisfying the assumptions (K)–(O) be  = ξ  (t), t ≥ 0 , κ  , η  , i ∈ X constructed as probability copies of the triplet ξ¯ε,i ε,i ε,i ε,i and the random variable ηε . The multi-alternating regenerative process (ξε (t), ηε (t)), t ≥ 0, built using the     above triplets ξ¯ε,i,n = ξε,i,n (t), t ≥ 0 , κε,i,n , ηε,i,n

, i ∈ X, n = 1, 2, . . ., the random  variable ηε , and the relations (10.1)–(10.3), is a probabilistic copy of the process (ξ˜ε (t), η˜ε (t)), t ≥ 0. This means that these processes have the same finite-dimensional distributions, i.e., (ξ˜ε (t), η˜ε (t)), t ≥ 0 = (ξε (t), ηε (t)), t ≥ 0. d

(10.67)

The above relation holds for any distribution of the random variable ηε = ηε (0), in particular, in the cases when P{ηε = ηε (0) = i} = 1, for i ∈ X. 10.2.1.2 Perturbation Conditions for Modulating Semi-Markov Process with Totally Removed Virtual Transitions. Transition probabilities for the Markov renewal process (η˜ε,n, κ˜ε,n ) take the following form:

10.2 MARP with removed virtual transitions

203

Q˜ ε,i j (t) = P{η˜ε,1 = j, κ˜ε,1 ≤ t/η˜ε,0 = i} ∞

= Q(∗n) ε,ii (t) ∗ Q ε,i j (t), t ≥ 0, i, j ∈ X,

(10.68)

n=0

where, as usual, Q(∗0) ε,ii (t) = I(t ≥ 0), t ≥ 0. The transition probabilities for the embedded Markov chain η˜ ε,n are given by the following relation: p˜ε,i j = P{η˜ ε,1 = j/η˜ ε,0 = i} pε,i j = I( j  i) , i, j ∈ X, p¯ε,ii where p¯ε,ii = 1 − pε,ii =



pε,i j , i ∈ X.

(10.69)

(10.70)

ji

Note that the condition H1 implies that the probabilities p¯ε,ii = 1−pε,ii > 0, i ∈ X, for ε ∈ (0, 1]. Let us also introduce the distribution functions of sojourn time for the semiMarkov process η˜ε (t),

F˜ε,i (t) = (10.71) Q˜ ε,i j (t), t ≥ 0. j ∈X

In Sect. 9.21 , the procedure is described for transforming the conditions G, H, I, IH , L, LH , J, J◦ , and K for the semi-Markov processes ηε (t) in similar conditions ˜ H, ˜ I, ˜ I˜ H , L, ˜ L ˜ H , J, ˜ J˜ ◦ , and K ˜ for the semi-Markov processes η˜ε (t). G, In the same way, the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , J◦1 , and K1 can be ˜ 1, H ˜ 1 , I˜ 1 , L ˜ H,1 , L ˜ 1 , I˜ H,1 , J˜ 1 , J˜ ◦ , and K ˜ 1. transformed in similar conditions G 1 Analogues of the conditions G1 and H1 take for the semi-Markov processes η˜ε (t) the following forms: ˜ 1 : (a) p˜ε,i j > 0, ε ∈ (0, 1] or p˜ε,i j = 0, ε ∈ (0, 1], for every i, j ∈ X, (b) F˜ε,i (0) = 0, G for i ∈ X, ε ∈ (0, 1]. and ˜ 1 : For any i, j ∈ X, there exist n˜ i j ≥ 1 and a chain of states i = r˜0, r˜1, . . . , r˜n˜ i j = j H  such that 1≤l ≤ n˜ i j p˜1, r˜l−1 r˜l > 0. The following lemma is a direct corollary of Lemmas 9.31 and 9.51 . Lemma 10.4 The conditions G1 and H1 , assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of the conditions G1 and H1 for the semi-Markov ˜ 1 and H ˜ 1. processes η˜ε (t) in the form of conditions G For i ∈ X and ε ∈ (0, 1], we introduce the sets, ˜ ε,i = { j ∈ X : p˜ε,i j > 0}. Y The following lemma is an analogue of Lemma 9.41 .

(10.72)

204

10 Perturbed MARP

Lemma 10.5 Let the conditions G1 and H1 be satisfied for semi-Markov processes ˜ ε,i = Y ˜ 1,i, ε ∈ (0, 1], for i ∈ X and, ηε (t). Then, the sets Y ˜ 1,i = Y1,i \ {i}, for i ∈ X. Y

(10.73)

Analogues of the conditions I˜ 1 and IH,1 take for the semi-Markov processes η˜ε (t) the following form: I˜ 1 : p˜ε,i j → p˜0,i j as ε → 0, for i, j ∈ X and ˜ 1,i, i ∈ X belong to the complete family of asymptotI˜ H,1 : The functions p˜ ·,i j , j ∈ Y ically comparable functions H appearing in the condition IH,1 . Obviously,  p˜0,i j  is a stochastic matrix. Note also that the condition I˜ 1 is implied by the condition I˜ H,1 . Let us introduce the set of asymptotically absorbing states, Y0 = {i ∈ X : p0,ii = 1}.

(10.74)

The following lemma is a direct corollary of Lemma 9.71 . Lemma 10.6 The conditions G1 , H1 , I1 , and IH,1 , assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of the conditions I1 and IH,1 for the semi-Markov processes η˜ε (t) in the form of conditions I˜ 1 and I˜ H,1 . The limiting transition probabilities p˜0,i j appearing in the condition I˜ 1 take the following form: p ¯ 0, j ∈ X, I( j  i) p¯0,0, iiij for i ∈ Y (10.75) p˜0,i j = pε, i j I( j  i) limε→0 p¯ ε, ii for i ∈ Y0, j ∈ X.  Note that the condition IH,1 implies that the functions p ·,i j and pε,ii = ki p ·,ik belong to the family H and, thus, the limits in the relation (10.75) exist. The limiting probabilities p˜0,i j appearing in the condition I˜ 1 satisfy the following relations:

p˜0,i j ≥ 0, j  i, p˜0,ii = 0, i ∈ X, and p˜0,i j = 1, i ∈ X. (10.76) ji

The normalisation functions u˜ε,i, i ∈ X used in the condition J˜ 1 formulated below take the following forms, for i ∈ X: −1 u˜ε,i = p¯−1 ε,ii uε,i = (1 − pε,ii ) uε,i .

(10.77)

Note that from conditions G1 and H1 it follows that p¯ε,ii ∈ (0, 1], ε ∈ 0, 1], for i ∈ X and, thus, by the condition J1 (c), for i ∈ X, u˜ε,i ∈ (0, ∞), ε ∈ (0, 1].

(10.78)

10.2 MARP with removed virtual transitions

205

Analogues of the conditions L1 and LH,1 take for the semi-Markov processes η˜ε (t) the following forms: ˜ 1 : u˜ε,i → u˜0,i ∈ (0, ∞] as ε → 0, for i ∈ X. L and ˜ H,1 : The functions u˜ ·,i, i ∈ X belong to the complete family of asymptotically L comparable functions H appearing in the condition LH,1 . The following lemma is a direct corollary of Lemma 9.121 . Lemma 10.7 The conditions G1 , H1 and I1 , IH,1 , L1 , LH,1 , assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of the condition L1 and ˜ 1 and L ˜ H,1 . LH,1 for the semi-Markov processes η˜ε (t) in the form of conditions L Note, also, that the conditions G1 , H1 , I1 , L1 and the relation (10.78) imply that ˜ 2 take the following forms: the limits u˜0,i, i ∈ X appearing in the condition L  −1 p¯ u0,i ∈ (0, ∞) if p¯0,ii > 0, u0,i ∈ (0, ∞), u˜0,i = 0,ii (10.79) ∞ if p¯0,ii = 0 or p¯0,ii > 0, u0,i = ∞. The corresponding distribution function F˜ε,i j (u) and its Laplace transform φ˜ε,i j (s) ˜ 1,i, i ∈ X: take the following forms, for ε ∈ (0, 1] and j ∈ Y F˜ε,i j (t) = P{ κ˜ε,1 ≤ t/η˜ε,0 = i, η˜ε,1 = j} ∞ 1 (∗n) n = F (t) ∗ Fε,i j (t)pε,ii pε,i j , t ≥ 0, p˜ε,i j n=0 ε,ii ∫

and φ˜ε,i j (s) =

0

∞

e−su F˜ε,i j (du) =

φε,i j (s) p¯ε,ii , s ≥ 0. 1 − φε,ii (s)pε,ii

(10.80)

(10.81)

Also, we can define the corresponding distribution function F˜ε,i j (u) and its ˜ 1,i, i ∈ X, Laplace transform φ˜ε,i j (s), for ε ∈ (0, 1] and j  Y F˜ε,i j (t) = F˜ε,i (t), t ≥ 0 ∫

and φ˜ε,i j (s) = φ˜ε,i (s) =

0

∞

e−su F˜ε,i (du), s ≥ 0.

(10.82)

(10.83)

¯ 1,i, i ∈ X does not affect the Note that the above choice of F˜ε,i j (t) for j ∈ Y ˜ transition probabilities Q ε,i j (t) of the semi-Markov process η˜ε (t) and analogues of the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , J◦1 , and K1 for the semi-Markov processes η˜ε (t) (see, Sect. 10.1.1.1 and Remark 10.1). Analogues of the conditions J1 and J◦1 take for the semi-Markov processes η˜ε (t) the following forms:

206

10 Perturbed MARP

˜ 1,i, i ∈ X, (b) F˜0,i j (·) is a nonJ˜ 1 : (a) F˜ε,i j (· u˜ε,i ) ⇒ F˜0,i j (·) as ε → 0, for j ∈ Y ˜ 1,i, i ∈ X, arithmetic distribution function without singular component, for j ∈ Y (c) u˜ε,i ∈ (0, ∞), ε ∈ (0, 1], for i ∈ X and ˜ 1,i, i ∈ X, (b) φ˜0,i j (·) is J˜ ◦1 : (a) φ˜ε,i j (s/u˜ε,i ) → φ˜0,i j (s) as ε → 0, for s ≥ 0 and j ∈ Y the Laplace transforms of a non-arithmetic distribution function without singular ˜ 1,i, i ∈ X, (c) u˜ε,i ∈ (0, ∞), ε ∈ (0, 1], for i ∈ X. component F˜0,i j (·), for j ∈ Y Below, we shall prove that the normalisation functions u˜ ·,i, i ∈ X defined in the relation (10.77) can be used in the conditions J˜ 1 and J˜ ◦1 . ˜ 1, H ˜ 1 , and I˜ 1 imply that, for i ∈ X, The conditions G ˜ 0,i = { j ∈ X : p0,i j > 0} ⊆ Y ˜ 1,i . Y

(10.84)

˜ 1, H ˜ 1 , I˜ 1 , J˜ 1 , and the relation (10.84) imply that, for i ∈ X, The conditions G



Q˜ ε,i j (·u˜ε,i ) = F˜ε,i j (·u˜ε,i ) p˜ε,i j F˜ε,i (·u˜ε,i ) = ˜ 1, i j ∈Y

⇒



˜ 1, i j ∈Y

F˜0,i j (·) p˜0,i j =

˜ 1, i j ∈Y



F˜0,i j (·) p˜0,i j

˜ 0, i j ∈Y

= F˜0,i (·) as ε → 0.

(10.85)

˜ 1, H ˜ 1 , I˜ 1 , J˜ 1 , and the relation (10.84) imply that, for i, j ∈ X, The conditions G Q˜ ε,i j (·u˜ε,i ) = F˜ε,i j (·u˜ε,i ) p˜ε,i j ⇒ F˜0,i j (·) p˜0,i j = Q˜ 0,i j (·) as ε → 0. Note that,

 ˜ 0,i, i ∈ X, F˜0,i j (·) p˜0,i j for j ∈ Y Q˜ 0,i j (·) = ¯ ˜ 0 for j ∈ Y0,i, i ∈ X.

(10.86)

(10.87)

The following lemma is an analogue of Lemma 9.101 . Lemma 10.8 The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J◦1 (J1 ), and K1 , assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of the condition J◦1 (J1 ) for the semi-Markov processes η˜ε (t) in the form of condition J˜ ◦1 (J˜ 1 ), where: ˜ 1,i, i ∈ X appearing in the (i) The limiting Laplace transforms φ˜0,i j (·), j ∈ Y ◦ ˜ condition J1 take two different forms given by the relation (10.88), for cases where ¯ 0 or j ∈ Y ˜ 1,i, i ∈ Y0,. ˜ 1,i, i ∈ Y j∈Y (ii) The normalisation functions u˜ ·,i, i ∈ X appearing in the condition J˜ ◦1 (J˜ 1 ) are given by the relation (10.77). Proof Lemma 9.101 implies the fulfilment of the asymptotic relation given in the condition J˜ ◦1 , with the corresponding limiting Laplace transforms φ˜0,i j (·) given by the

10.2 MARP with removed virtual transitions

207

relations (9.52)1 , (9.53)1 , and (9.63)1 . This asymptotic relation takes the following form: φ˜ε,i j (s/u˜ε,i ) → φ˜0,i j (s) φ ( p¯ =

¯ 0, ii 0, i j 0, ii s) p 1−p0, ii φ0, ii ( p¯ 0, ii s) 1 1+s f0, ii

¯ 0, j ∈ Y ˜ 1,i, for s ≥ 0, i ∈ Y ˜ 1,i . for s ≥ 0, i ∈ Y0, j ∈ Y

(10.88)

These relation can be re-written in the equivalent form using characteristic functions instead of Laplace transform. The corresponding limiting characteristic functions ϕ˜0,i j (z) of the limiting distribution functions F˜0,i j (·) take, in this case, the following form: ϕ ( p¯ z) p¯ 0, i j 0, ii 0, ii ˜ 1,i, i ∈ Y ¯ 0, for z ∈ R1, j ∈ Y (10.89) ϕ˜0,i j (z) = 1−p10, ii ϕ0, ii ( p¯0, ii z) ˜ for z ∈ R1, j ∈ Y1,i, i ∈ Y0 . 1−iz f0, ii According to the condition J1 , the distribution functions F0,i j (·), j ∈ Y1,i, i ∈ X are assumed to be non-arithmetic, and, thus, by Lemma A.1, |ϕ0,i j (z)| < 1, z  0, ¯ 0. ˜ 1,i, i ∈ X. Also, the probabilities p¯0,ii > 0, for i ∈ Y for j ∈ Y ¯ 0, ˜ 1,i, i ∈ Y Therefore, for j ∈ Y |ϕ0,i j ( p¯0,ii z)| p¯0,ii 1 − p0,ii |ϕ0,ii ( p¯0,ii z)| ≤ |ϕ0,i j ( p¯0,ii z)| < 1, for z  0.

| ϕ˜0,i j (z)| ≤

(10.90)

˜ 1,i, i ∈ Y ¯ 0 are nonThus, by Lemma A.1, the distribution function F˜0,i j (·), j ∈ Y arithmetic. Also, pε,ii → p0,ii = 1 as ε → 0 and, thus, state i ∈ Y1,i , for i ∈ Y0 . Therefore, by conditions J1 and K1 , the limiting expectations f0,ii > 0, for i ∈ Y0 . According to relation (10.89), the distribution function F˜0,i j (·) is exponential with ˜ 1,i, i ∈ Y0 , and, thus, it is non-arithmetic. parameter f0,ii > 0, for j ∈ Y ˜ 1,i, i ∈ X are nonBy the above remarks, all distribution functions F˜0,i j (·), j ∈ Y arithmetic. ˜ 1,i, i ∈ Y ¯ 0 , and, The relation (10.88) implies that, p¯0,ii > 0, for j ∈ Y F0,i j (·) if p¯0,ii = 1, (10.91) F˜0,i j (·) = ∞ (∗n) −1 ) ∗ F −1 )pn p¯ F (· p ¯ (· p ¯ if p¯0,ii < 1. 0,i j 0,ii n=0 0,ii 0,ii 0,ii 0,ii ˜ 1,i = Y1,i \ {i}, for i ∈ X. In this case, according to the condition J1 , the The set Y ˜ 1,i, i ∈ X have no singular component. distribution functions F0,i j (·), j ∈ Y Also, if p¯0,ii < 1, then i ∈ Y1,i and, according to the condition J1 , the distribution F0,ii (·) has no singular component.

208

10 Perturbed MARP

The above remarks and the relation (10.91) imply that the distribution functions ˜ 1,i, i ∈ Y ¯ 0 (in this case, 0 < p¯0,ii ≤ 1) have no singular component. F˜0,i j (·), j ∈ Y ˜ 1,i, i ∈ Y0 (in this case, p¯0,ii = 0) are Also, the distribution function F˜0,i j (·, ) j ∈ Y exponential and, thus, have no singular component. Thus, the condition J1 holds for the semi-Markov processes η˜ε (t) in the modified ˜ 1,i, i ∈ X form of the condition J˜ 1 , where the assumption, F˜0,i j (0) < 1, j ∈ Y is replaced by the stronger assumption that the limiting distribution functions ˜ 1,i, i ∈ X are non-arithmetic and have no singular component.  F˜0,i j (·), j ∈ Y The corresponding expectations f˜ε,i j are given by relation (9.65)1 and take the ˜ 1,i, i ∈ X and ε ∈ (0, 1]: following forms, for j ∈ Y ∫ ∞ −1 ˜fε,i j = u F˜ε,i j (du) = fε,i j + pε,ii p¯ε,ii fε,ii . (10.92) 0

An analogue of the condition K1 takes for the semi-Markov processes η˜ε (t) the following form: ˜ 1,i, i ∈ X, for every ε ∈ (0, 1], (b) f˜ε,i j /u˜ε,i → f˜0,i j = ˜ 1 : (a) f˜ε,i j < ∞, j ∈ Y K ∫∞ ˜ 1,i, i ∈ X. t F˜0,i j (dt) < ∞ as ε → 0, for j ∈ Y 0 ˜ 1, H ˜ 1 , I˜ 1 , J˜ 1 , K ˜ 1 , and relation (10.84) imply that, for i ∈ X, The conditions G

f˜ε,i j f˜ε,i = p˜ε,i j u˜ε,i u˜ ˜ 1, i ε,i j ∈Y

→ f˜0,i j p˜0,i j ˜ 1, i j ∈Y

=



f˜0,i j p˜0,i j = f˜0,i as ε → 0.

(10.93)

˜ 0, i j ∈Y

˜ 1, H ˜ 1 , I˜ 1 , J˜ 1 , K ˜ 1 and the relations (10.84) and (10.93) Also, the conditions G imply that, for i, j ∈ X, f˜ε,i j e˜ε,i j = p˜ε,i j → f˜0,i j p˜0,i j = e˜0,i j as ε → 0. u˜ε,i u˜ε,i Note that,

 e˜0,i j =

˜ 0,i, i ∈ X, f˜0,i j p˜0,i j for j ∈ Y ˜ 0,i, i ∈ X. 0 for j  Y

(10.94)

(10.95)

The above remarks can be summarised in the following lemma, which is a direct corollary of Lemma 9.111 . Lemma 10.9 The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , and K1 , assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of the condition ˜ 1 , where: K1 for the semi-Markov processes η˜ε (t) in the form of the condition K

10.2 MARP with removed virtual transitions

209

˜ 1,i, i ∈ X appearing in condition K ˜ 1 are (i) The limiting expectations f˜0,i j , j ∈ Y given by the relation (10.96). ˜ 1 are given (ii) The normalisation functions u˜ ·,i, i ∈ X appearing in condition K by the relation (10.77). Proof Lemma 9.111 implies holding of the asymptotic relation given in condition ˜ 1 , with the corresponding limits f˜0,i j , which are given by the relation (9.67)1 . This K ˜ 1,i, i ∈ X: asymptotic relation takes the following form, for j ∈ Y fε,i j f˜ε,i j fε,ii = p¯ε,ii + pε,ii u˜ε,i uε,i uε,i → p¯0,ii f0,i j + p0,ii f0,ii = f˜0,i j as ε → 0.

(10.96)

˜ 1,i, i ∈ Y ¯0 The above formula for the limits f˜0,i j works for both cases, where j ∈ Y ˜ or j ∈ Y1,i, i ∈ Y0 .  10.2.1.3 The Perturbation Condition R for Multi-Alternating Regenerative Processes with Totally Removed Virtual Transitions for Modulating SemiMarkov Process. The form of the functions qε,i (tuε,i, A) used in the condition R suggests that the role of these functions for the multi-alternating regenerative processes (ξ˜ε (t), η˜ε (t)) can execute functions, q˜ε,i (t u˜ε,i, A), t ≥ 0, i ∈ X, A ∈ BZ,

(10.97)

where the normalisation functions uε,i, i ∈ X are given by the relation (10.77) and, q˜ε,i (t, A) = Pi { ξ˜ε (t) ∈ A, ζ˜ε,1 > t}, t ≥ 0, i ∈ X, A ∈ BZ .

(10.98)

The relation (10.98) implies that the functions q˜ε,i (t u˜ε,i, A), t ∈ R+, A ∈ BZ belong to the class P[BZ ], for i ∈ X, ε ∈ (0, 1]. Moreover, the function q˜ε,i (t u˜ε,i, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − F˜ε,i (t u˜ε,i ), for i ∈ X and ε ∈ (0, 1], i.e., q˜ε,i (t u˜ε,i, Z) = 1 − F˜ε,i (t u˜ε,i ), for t ∈ R+ .

(10.99)

An analogue of the condition R takes for the multi-alternating regenerative processes (ξ˜ε (t), η˜ε (t)) the following form: ˜ There exist functions q˜0,i (t, A), t ≥ 0, A ∈ BZ , i ∈ X, which belong to the class R: P[BZ ], a class of set Γ ⊆ BZ , and Borel sets U[q˜ ·,i (· u˜ ·,i, A)], A ∈ Γ, i ∈ X such that: (a) the function q˜0,i (t, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − F˜0,i (t), t ∈ R+ , for i ∈ X; (b) the functions us q˜ε,i (· u˜ε,i, A) −→ q˜0,i (·, A) as ε → 0, for points s ∈ U[q˜ ·,i (· u˜ ·,i, A)], A ∈ Γ, i ∈ X; (c) m(U¯ [q˜ ·,i (· u˜ ·,i, A)]) = 0, for A ∈ Γ, i ∈ X; (d) the function q˜0,i (·, A) is continuous almost everywhere with respect to the Lebesgue measure m(·) on B+ , for A ∈ Γ, i ∈ X.

210

10 Perturbed MARP

It should be noted that the notation U[q˜ ·,i (· u˜ ·,i, A)] is used to indicate that this set of convergence is determined by the family of functions q˜ε,i (· u˜ε,i, A), ε ∈ (0, 1]. It is also helpful to make a few comments about the above perturbation condition. ˜ (a) is natural. In light of the relation (10.99), the consistency condition R The consistency relation (10.99) implies that, for every ε ∈ (0, 1] and A ∈ BZ, i ∈ X, the function q˜ε,i (·u˜ε,i, A) is majorised by the tail probability function 1− F˜ε,i (·u˜ε,i ) on the interval [0, ∞), i.e., q˜ε,i (t u˜ε,i, A) ≤ 1 − F˜ε,i (t u˜ε,i ), for t ∈ R+ .

(10.100)

˜ (a) implies that the following majorisation relation similar to The condition R (10.100) takes place, for A ∈ BZ, i ∈ X: q˜0,i (t, A) ≤ 1 − F˜0,i (t), for t ∈ R+ .

(10.101)

˜ 1,i, i ∈ X Note also that the choice of the distribution functions F˜ε,i j (·), j  Y ˜ according to the relation (10.82) does not affect the condition R since the functions q˜ε,i (·, A), A ∈ BZ, i ∈ X do not depend on the above distribution functions. The following lemma takes place. Lemma 10.10 The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , and R, assumed to be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)), entail the fulfilment of the condition R for the multi-alternating regenerative processes ˜ where: (ξ˜ε (t), η˜ε (t)) in the form of condition R, ˜ is defined for (i) The limiting function q˜0,i (·, A) appearing in the condition R A ∈ BZ, i ∈ X by the relation (10.126), if p¯0,ii = 1, or by the relation (10.135), if p¯0,ii ∈ (0, 1), or by the relation (10.168), if p¯0,ii = 0. ˜ coincides with the class of (ii) The class of sets Γ appearing in the condition R sets Γ appearing in the condition R. ˜ is defined (iii) The set of convergence U[q˜ ·,i (· u˜ ·,i, A)] appearing in the condition R for A ∈ Γ, i ∈ X by the relation (10.127), if p¯0,ii = 1, or by the relation (10.158), if p¯0,ii ∈ (0, 1), or by the relation (10.169), if p¯0,ii = 0. ˜ are given (iv) The normalisation functions u˜ ·,i, i ∈ X appearing in the condition R by the relation (10.77). Proof For each i ∈ X and ε ∈ [0, 1], we introduce a sequence of stochastic triplets, ξ¯ε,i,n = ξε,i,n (t), t ≥ 0 , κε,i,n, με,i,n , n = 1, 2, . . . ,

(10.102)

where με,i,n = κε,i,n I(ηε,i,n = i), n = 1, 2, . . . .

(10.103)

Also, let us define for each ε ∈ [0, 1] a regenerative process, ξε,i (t) = ξε,i,n (t − ζε,i,n−1 ), for t ∈ [ζε,i,n−1, ζε,i,n ), n = 1, 2, . . . , with regeneration times,

(10.104)

10.2 MARP with removed virtual transitions

211

ζε,i,n = κε,i,1 + · · · + κε,i,n, n = 1, 2, . . . , ζε,i,0 = 0,

(10.105)

and a regenerative lifetime, με,i,+ = ζε,i,νε, i ,

(10.106)

where, νε,i = min(n ≥ 1 : με,i,n < κε,i,n ) = min(n ≥ 1 : ηε,i,n  i).

(10.107)

Let us introduce the probabilities, for A ∈ Γ, t ≥ 0, Pε,i,+ (t, A) = P{ξε,i (t) ∈ A, με,i,+ > t}. In this case, the distribution functions of regeneration times,

Fε,i (u) = P{κε,i,1 ≤ u} = Q ε,i j (u), u ≥ 0,

(10.108)

(10.109)

j ∈X

and Fε,i,+ (u) = P{κε,i,1 ≤ u, με,i,1 ≥ κε,i,1 } = P{κε,i,1 ≤ u, ηε,i,1 = i} = Q ε,ii (u), u ≥ 0.

(10.110)

The function Pε,i,+ (t, A), t ≥ 0 is for any i ∈ X, A ∈ BZ and ε ∈ (0, 1], the only solution from the class L for the following renewal equation: ∫ t Pε,i,+ (t, A) = qε,i (t, A) + Pε,i,+ (t − u, A)Fε,i,+ (du), t ≥ 0, (10.111) 0

where qε,i (t, A) = P{ξε,i (t) ∈ A, ζε,i,1 ∧ με,i,+ > t} = P{ξε,i (t) ∈ A, ζε,i,1 > t}, t ≥ 0.

(10.112)

It is used here that P{με,i,+ ≥ ζε,i,1 } = 1 according to the relations (10.106) and (10.107). Note that Fε,i,+ (·) is generally an improper distribution function, and, Fε,i,+ (∞) = Q ε,ii (∞) = pε,ii .

(10.113)

The corresponding stopping probability, qε,i,+ = P{με,i,1 < κε,i,1 } = P{ηε,i,1  i} = p¯ε,ii =

ji

Also the expectations of inter-regeneration times,

pε,i j .

(10.114)

212

10 Perturbed MARP

eε,i = E κε,i,1 =



eε,i j ,

(10.115)

j ∈X

where eε,i j = E κε,i,1 I(ηε,i,1 = j), j ∈ X,

(10.116)

eε,i,+ = E κε,i,1 I(με,i,1 ≥ κε,i,1 ) = E κε,i,1 I(ηε,i,1 = i) = eε,ii .

(10.117)

and

The solution of the renewal equation (10.111) has the following form, for i ∈ X, A ∈ BZ and ε ∈ (0, 1]: ∫ t qε,i (t − s, A)Uε,i,+ (ds), t ≥ 0, (10.118) Pε,i,+ (t, A) = 0

where the renewal function Uε,i,+ (·) takes the following form, for i ∈ X: Uε,i,+ (s) =

∞

(∗n) Fε,i,+ (s), s ≥ 0.

(10.119)

n=0

The following relation, which holds for any t ≥ 0, i ∈ X, A ∈ BZ , plays a key role in what follows: q˜ε,i (t u˜ε,i, A) = Pi { ξ˜ε (t u˜ε,i ) ∈ A, ζ˜ε,1 > t u˜ε,i } = P{ξε,i (t u˜ε,i ) ∈ A, με,i,+ > t u˜ε,i } = Pε,i,+ (t u˜ε,i, A).

(10.120)

The function Pε,i,+ (t u˜ε,i, A) = q˜ε,i (t u˜ε,i, A), t ≥ 0 is for any i ∈ X, A ∈ BZ and ε ∈ (0, 1] the only solution from the class L for the following renewal equation: Pε,i,+ (t u˜ε,i, A) = qε,i (t u˜ε,i, A) ∫ t P˜ε,i,+ ((t − s)u˜ε,i, A)Fε,i,+ (u˜ε,i ds), t ≥ 0. +

(10.121)

0

The solution of the above renewal equation has the following form, for t ≥ 0: Pε,i,+ (t u˜ε,i, A) = q˜ε,i (t u˜ε,i, A) ∫ t qε,i ((t − s)u˜ε,i, A)Uε,i,+ (u˜ε,i ds), =

(10.122)

0

where the renewal function Uε,i,+ (u˜ε,i ·) takes the following form, for i ∈ X: Uε,i,+ (u˜ε,i s) =

∞

n=0

(∗n) Fε,i,+ (u˜ε,i s), s ≥ 0.

(10.123)

10.2 MARP with removed virtual transitions

213

We are now ready to perform a detailed asymptotic analysis for the functions q˜ε,i (t u˜ε,i, A). The condition I1 implies that qε,i = p¯ε,ii → q0,i = p¯0,ii as ε → 0.

(10.124)

For each i ∈ X, three possible cases should be considered: (1) p¯0,ii = 1, (2) p¯0,ii ∈ (0, 1), and (3) p¯0,ii = 0. First, let us assume that: (1) p¯0,ii = 1. In this case, according to the condition I1 , u˜ε,i = p¯−1 ε,ii → 1 as ε → 0. uε,i

(10.125)

˜ holds, with the limiting functions q˜0,i (t, A), t ≥ Let us prove that the condition R 0, A ∈ BZ given by the following relation: q˜0,i (t, A) = q0,i (t, A), t ≥ 0, A ∈ BZ

(10.126)

and the corresponding sets of locally uniform convergence for the functions q˜ε,i (· u˜ε,i, A), A ∈ Γ given by the following relation: U[q˜ ·,i (· u˜ ·,i, A)] = U[q ·,i (·u ·,i, A)].

(10.127)

Here, Γ is the class of sets appearing in the condition R. Since, pε,ii → 0 as ε → 0, the following relation holds, for s ≥ 0: 0 ≤ Uε,i,+ (u˜ε,i s) − I(u˜ε,i s ≥ 0) ∞ ∞



(∗n) n ≤ Fε,i,+ (u˜ε,i s) ≤ Fε,i,+ (u˜ε,i s) n=1

≤

∞

n=1

n Fε,i,+ (∞) =

n=1

∞

n pε,ii =

n=1

pε,ii → 0 as ε → 0. 1 − pε,ii

(10.128)

The relations (10.122) and (10.128) imply that, for t ≥ 0, A ∈ BZ and ε ∈ (0, 1], 0 ≤ q˜ε,i (t u˜ε,i, A) − qε,i (t u˜ε,i, A) ∫ t ∞

(∗n) = qε,i (u˜ε,i (t − s), A)( Fε,i,+ (u˜ε,i ds)) 0

≤

∞

n=1

n=1 (∗n) Fε,i,+ (u˜ε,i t) ≤

pε,ii . 1 − pε,ii

The relation (10.129) implies that, for A ∈ BZ and s ≥ 0,

(10.129)

214

10 Perturbed MARP us

q˜ε,i (·u˜ε,i, A) − qε,i (·u˜ε,i, A) −→ 0(·) ≡ 0 as ε → 0.

(10.130)

Also, the asymptotic relation given in the condition R (b) and the relation (10.125) imply that the following relation holds, for A ∈ Γ and s ∈ U[q ·,i (·u ·,i, A)]: qε,i (·u˜ε,i, A) = qε,i (·uε,i

u˜ε,i us , A) −→ q0,i (· , A) as ε → 0. uε,i

(10.131)

The relations (10.130) and (10.131) imply that, for A ∈ Γ and s ∈ U[q ·,i (·u ·,i, A)], us

q˜ε,i (·u˜ε,i, A) −→ q0,i (· , A) as ε → 0.

(10.132)

Thus, we can choose the function q˜0,i (t, A) = q0,i (t, A), t ∈ R+, A ∈ BZ , the class Γ is the same as in the condition R, and the sets U[q˜ ·,i (· u˜ ·,i, A)] = U[q ·,i (·u ·,i, A)], A ∈ Γ. According to the condition R (a), the function q0,i (t, A), t ≥ 0, A ∈ BZ belongs to class P[Z] and is consistent with the tail probability function 1 − F0,i (t), t ≥ 0, ˜ (a) is satisfied. The relation (10.132) and the condition for i ∈ X. Thus, condition R ˜ (b) and (c) are satisfied. R (c) imply that the relations given in the conditions R Finally, the condition R (d) implies that the continuity assumption formulated in the ˜ (d) also holds for the functions q0,i (·, A), A ∈ Γ. condition R Second, let us assume that: (2) p¯0,ii ∈ (0, 1). According to the condition I1 , the probabilities pε,ii → p0,ii ∈ (0, 1) as ε → 0 and, thus, u˜ε,i −1 = p¯−1 (10.133) ε,ii → p¯0,ii ∈ (0, ∞) as ε → 0. uε,i In this case, the state i ∈ Y1,i and, thus, the conditions G1 , I1 , J1 and the relations (10.17) and (10.110) imply that, Fε,i,+ (·u˜ε,i ) = Q ε,ii (·u˜ε,i ) u˜ε,i )pε,ii ⇒ F0,ii (· p¯−1 = Fε,ii (·uε,i 0,ii )p0,ii uε,i −1 = Q0,ii (· p¯−1 0,ii ) = F0,i,+ (· p¯0,ii ) as ε → 0.

(10.134)

˜ is satisfied, with the limiting functions Let us prove that, the condition R q˜0,i (t, A), t ≥ 0, A ∈ BZ given by the following relation: ∫ t −1 q0,i ((t − s) p¯−1 (10.135) q˜0,i (t, A) = 0,ii, A)U0,i,+ ( p¯0,ii ds), A ∈ BZ, t ≥ 0, 0

where, for i ∈ X, U0,i,+ (s) =

∞

n=0

(∗n) F0,i,+ (s) =

∞

n=0

Q(∗n) 0,ii (s), s ≥ 0,

(10.136)

10.2 MARP with removed virtual transitions

215

and the corresponding sets of locally uniform convergence U[q˜0,i (·u˜ ·,i, A)], A ∈ Γ for the functions q˜ε,i (·, A) are given by the relation (10.158). Here, as above, Γ is the class of sets appearing in the condition R. The function q˜0,i (t, A), t ∈ R+, A ∈ BZ , given by the relation (10.135), belongs to the class P[Z], since the function q0,i (t, A), t ∈ R+, A ∈ BZ belongs to the class P[Z] according to the condition R. ˜ (a) is satisfied, that is, the function q˜0,i (t, A), t ∈ Let us check that the condition R R+, A ∈ BZ is consistent with the tail probability functions 1 − F˜0,i (t), t ∈ R+ , given by relations (10.85). The relations (10.85), (10.88), and (10.91) imply that, for t ≥ 0,

F˜0,i j (·) p˜0,i j F˜0,i (·) = j ∈X ∞



=

(∗n) −1 n F0,ii (· p¯−1 0,ii t) ∗ F0,i j (· p¯0,ii )p0,ii p¯0,ii

ji n=0



=

ji t

−1 Q0,i j ( p¯−1 0,ii t) ∗ U0,i,+ ( p¯0,ii t)

∫ =

0

p0,i j p¯0,ii

(



−1 Q0,i j ( p¯−1 0,ii (t − s)))U0,i,+ ( p¯0,ii ds).

(10.137)

ji

Also, the condition R (a) and the relations (10.135) and (10.137) imply that, for t ≥ 0, ∫ t q0,i ((t − s, Z)u˜0,i, A)U0,i,+ ( p¯−1 q˜0,i (t, Z) = 0,ii ds) 0 ∫ t −1 (1 − F0,i ( p¯−1 = 0,ii (t − s))U0,,+i ( p¯0,ii ds) 0 ∫ t (1 − Q0,ii p¯−1 = 0,ii (t − s)) 0

−1 Q0,i j ( p¯−1 − 0,ii (t − s)) U0,i,+ ( p¯0,ii ds) ∫ = 0

ji t

−1 ˜ (1 − Q0,ii ( p¯−1 0,ii (t − s))U0,i,+ ( p¯0,ii ds) − F0,i (t)

= 1 − F˜0,i (t). The above relation uses that the function, ∫ t (1 − Q0,ii (u˜0,i (t − s))U0,i (u˜0,i ds), t ≥ 0, f0,i (t) = 0

is the only solution in the class L for the following renewal equation:

(10.138)

(10.139)

216

10 Perturbed MARP

∫ f0,i (t) = 1 − Q0,ii (u˜0,i t) +

t

f0,i (t − s)Q0,ii (u˜0,i ds), t ≥ 0.

(10.140)

0

On the other hand, it is easy to see that the solution to this equation is the function, f0,i (t) = 1, t ≥ 0.

(10.141)

˜ (b) and (c) are satisfied for the functions Let us now prove that the conditions R q˜ε,i (t u˜ε,i, A). The relation (10.113) and the above remarks imply that following relation takes place, for u ≥ 0: ∞

lim lim

N →∞ ε→0

(∗n) Fε,i,+ (uu˜ε,i ) ≤ lim lim

N →∞ ε→0

n=N

≤ lim lim

∞

N →∞ ε→0

= lim

n=N N p0,ii

N →∞

1 − p0,ii

n pε,ii

∞

n Fε,i,+ (uu˜ε,i )

n=N

= lim lim

N →∞ ε→0

N pε,ii

1 − pε,ii

= 0.

(10.142)

The relations (10.86), (10.142), and the above remarks imply that the following relation holds for renewal functions: Uε,i,+ (·u˜ε,i ) =

∞

(∗n) Fε,i,+ (·u˜ε,i ),

n=0 ∞

⇒

(∗n) −1 F0,i,+ (· p¯−1 0,ii ) = U0,i,+ (· p¯0,ii ) as ε → 0.

(10.143)

n=0

In the case (2), the probabilities p0,ii, p¯0,ii ∈ (0, 1). Thus, the renewal function U0,i,+ (· p¯−1 0,ii ) can be represented in the following form: −1 ˆ U0,i,+ (s p¯−1 0,ii ) = p¯0,ii F0,i (s),

where Fˆ0,i (s) = p¯−1 0,ii

∞

(∗n) n F0,ii (s p¯−1 0,ii )p0,ii p¯0,ii .

(10.144)

(10.145)

n=0

Obviously, Fˆ0,i (·) is a proper distribution function. By the condition J1 , the distribution function F0,ii (·) has no singular component. This, obviously, implies that the distribution function Fˆ0,i (·) also has no singular component, and, thus, it can be represented in the form, Fˆ0,i (·) = qi,a Fˆ0,i,a (·) + qi,d Fˆ0,i,d (·), where Fˆ0,i,a (·) is an absolutely continuous distribution function, Fˆ0,i,d (·) is a discrete distribution function, and qi,a, qi,d ≥ 0, qi,a + qi,d = 1.

10.2 MARP with removed virtual transitions

217

Let si,1, si,2, . . . be discontinuity points of the distribution function Fˆ0,i,d (s) and Si = {si,0 = 0, si,1, , si,2, . . .}. The set Si is a finite or countable set. Consider again the sets of convergence U[q ·,i (·u ·,i, A)], A ∈ Γ for the functions qε,i (·, A) appearing in the condition R. According to this condition, the Lebesgue measure m(U¯ [q ·,i (·u ·,i, A)]) = 0, for any A ∈ Γ. For A ∈ Γ and u > 0, we introduce sets, U[q ·,i (· u u ·,i, A)] = {s ≥ 0 : su ∈ U[q ·,i (· u ·,i, A)]},

(10.146)

¯ [q ·,i (· u u ·,i, A)] = {s ≥ 0 : su ∈ U¯ [q ·,i (· u ·,i, A)]}. U

(10.147)

and Obviously, m(U¯ [q ·,i (· u u ·,i, A)]) = 0.

(10.148)

The condition R and the relation (10.148) obviously imply that, for s ∈

U[q ·,i (· p¯−1 0,ii u ·,i, A)], A ∈ Γ,

us

qε,i (·u˜ε,i, A) −→ q0,i (· p¯−1 0,ii, A) as ε → 0.

(10.149)

For A ∈ Γ and ε ∈ [0, 1], we continue the function qε,i (t, A), t ≥ 0 as qε,i (t, A) = 0, for t < 0. For A ∈ Γ and u, v > 0, we introduce sets, U[q ·,i ((v − ·)u u ·,i, A)]

= {s ∈ [0, v] : (v − s) ∈ U[q ·,i (· u u ·,i, A)]} ∪ (v, ∞) = {s ∈ [0, v] : (v − s)u ∈ U[q ·,i (· u ·,i, A)]} ∪ (v, ∞),

(10.150)

and ¯ [q ·,i ((v − ·)u u ·,i, , A)] U = {s ∈ [0, v] : (v − s) ∈ U¯ [q ·,i (· u u ·,i, A)]} = {s ∈ [0, v] : (v − s)u ∈ U¯ [q ·,i (· u ·,i, A)].

(10.151)

Obviously, for A ∈ Γ and u, v > 0, m(U¯ [q ·,i ((v − ·)u u ·,i, A)]) = m(U¯ [q ·,i (·u u ·,i, A)] ∩ [0, v]) ≤ m(U¯ [q ·,i (·u u ·,i, A)]) = 0.

(10.152)

Let 0 ≤ tε → t > 0 as ε → 0. This relation, the condition R, and the relation (10.152) obviously imply that, for s ∈ U[q ·,i ((t − ·) p¯−1 0,ii u ·,i, A)], A ∈ Γ, us

qε,i ((tε − ·)u˜ε,i, A) −→ q0,i ((t − ·) p¯−1 0,ii, A) as ε → 0. We introduce sets, for A ∈ Γ,

(10.153)

218

10 Perturbed MARP

Ti, A = ∩si, n ∈Si {t > 0 : (t − si,n ) p¯−1 0,ii ∈ U[q ·,i (· u ·,i, A)]} −1 = ∩si, n ∈Si (U[q ·,i (· p¯−1 0,ii, A)]/ p¯0,ii + si,n ),

(10.154)

and ¯ T¯i, A = ∪si, n ∈Si {t > 0 : (t − si,n ) p¯−1 0,ii ∈ U[q ·,i (· u ·,i, A)]} −1 = ∪si, n ∈Si (U¯ [q ·,i (· u ·,i, , A)]/ p¯ + si,n ), 0,ii

(10.155)

where A/u + s denotes the set {a/u + s : a ∈ A}, for A ∈ B+, s ≥ 0, u > 0. For any set S ∈ B+ such that m(S) = 0 and s ≥ 0, u > 0, the set T = {t > 0 : (t − s)u ∈ S} has the Lebesgue measure m(T) = 0. Therefore, for A ∈ Γ, (10.156) m(T¯i, A) = 0. Also, if t ∈ Ti, A, then (t −sik ) p¯−1 0,ii ∈ U[q ·,i (· u ·,i, A)], for all si,n ∈ Si and, therefore, for any 0 ≤ tε → t as ε → 0. u s i, n

qε,i ((tε − ·)u˜ε,i, A) −→ q0,i ((t − ·) p¯−1 0,ii, A) as ε → 0.

(10.157)

By the definition of the sets T¯i, A, the probability Fˆ0,i,d (T¯i, A) = 0. Also, ˆ F0,i,a (T¯i, A) = 0, since m(T¯i, A) = 0. Therefore, Fˆ0,i (T¯i, A) = 0 and U0,i,+ (T¯i, A) = 0. Now we define sets of convergence as follows: U[q˜ ·,i (· u˜ ·,i, A)] = Ti, A, A ∈ Γ.

(10.158)

According to the relations (10.143), (10.157) and Lemma B.2, and above remarks, the following relation takes place, for t ∈ U[q˜ ·,i (· u˜ ·,i, A)], A ∈ Γ, and any 0 ≤ tε → t as ε → 0: ∫ tε qε,i ((tε − s)u˜ε,i, A)Uε,i,+ (u˜ε,i ds) q˜ε,i (tε u˜ε,i, A) = 0 ∫ ∞ = qε,i ((tε − s)u˜ε,i, A)Uε,i,+ (u˜ε,i ds) ∫0 ∞ −1 → q0,i ((t − s) p¯−1 0,ii, A)U0,i ( p¯0,ii ds) 0

= q˜0,i (t, A) as ε → 0.

(10.159)

The relation (10.159) implies that for any s ∈ U[q˜ ·,i (· u˜ ·,i, A)], A ∈ Γ, us

q˜ε,i (·u˜ε,i, A) −→ q˜0,i (·, A) as ε → 0.

(10.160)

˜ (d) is satisfied for the functions q˜0,i (·, A), A ∈ It remains to prove that condition R Γ, that is, these functions are continuous almost everywhere with respect to the Lebesgue measure on B+ .

10.2 MARP with removed virtual transitions

219

Let C[q0,i (·, A)] be the set of continuity points for the function q0,i (·, A), for A ∈ Γ. By the condition R (d), for A ∈ Γ, ¯ 0,i (·, A)]) = 0. m(C[q

(10.161)

We introduce sets, for A ∈ Γ, Ri, A = ∩si, n ∈Si {t > 0 : (t − si,n ) p¯−1 0,ii ∈ C[q0,i (·, A)]} = ∩si, n ∈Si (C[q0,i (·, A)]/ p¯−1 0,ii + si,n ),

(10.162)

and ¯ R¯i, A = ∪si, n ∈Si {t > 0 : (t − si,n ) p¯−1 0,ii ∈ C[q0,i (·, A)]} ¯ 0,i (·, A)]/ p¯−1 + si,n ). = ∪si, n ∈Si (C[q 0,ii

(10.163)

m( R¯i, A) = 0.

(10.164)

By the above remarks, for A ∈ Γ,

Also, if t ∈ Ri, A, then (t − si,n ) p¯−1 0,ii ∈ C[q0,i (·, A)], for all si,n ∈ Si and, therefore, for any 0 ≤ tε → t as ε → 0. q0,i ((tε − si,n ))u˜0,i, A) → q0,i ((t − si,n ))u˜0,i, A) as ε → 0.

(10.165)

From the definition of the sets R¯i, A, it follows that Fˆ0,i,d ( R¯i, A) = 0. Also, Fˆ0,i,a ( R¯i, A) = 0, since m( R¯i, A) = 0. Therefore, Fˆ0,i ( R¯i, A) = 0 and U0,i ( R¯i, A) = 0. According to the above remarks and the Lebesgue theorem, the following relation takes place, for t ∈ Ri, A, A ∈ Γ, and any 0 ≤ tε → t as ε → 0: ∫ tε −1 , A) = q0,i ((tε − s) p¯−1 q˜0,i (tε p¯−1 0,ii 0,ii, A)U0,i,+ ( p¯0,ii ds) 0 ∫ ∞ −1 = q0,i ((tε − s) p¯−1 0,ii, A)U0,i,+ ( p¯0,ii ds) 0 ∫ ∞ −1 → q0,i ((t − s) p¯−1 0,ii, A)U0,i,+ ( p¯0,ii ds) 0

= q˜0,i (t, A) as ε → 0.

(10.166)

As it was checked above, the function q˜0,i (t, A), t ≥ 0, A ∈ BZ belongs to the class P[Z] and is consistent with the tail probability function 1 − F˜0,i (t) = q˜0,i (t, Z), t ≥ 0. ˜ (a) is satisfied. Since, m(U¯ [q˜ ·,i (· u˜ ·,i, A)]) = 0, A ∈ Γ, the Thus, the condition R ˜ (b) and (c) are relation (10.160) implies that the relations given in the conditions R ˜ (d) satisfied. Finally, the relations (10.164) and (10.166) imply that the condition R is also satisfied for the functions q˜0,i (·, A), A ∈ Γ. Third, let us assume that: (3) p¯0,ii = 0.

220

10 Perturbed MARP

By the condition I1 , the probabilities pε,ii → p0,ii = 1 as ε → 0 and, thus, u˜ε,i = p¯−1 ε,ii uε,i → u˜0,i = ∞ as ε → 0.

(10.167)

˜ is satisfied, with the limiting functions Let us prove that the condition R q˜0,i (t, A), t ≥ 0, A ∈ BZ given by the following relation: q˜0,i (t, A) = e−t/e0, ii π0,i (A), t ≥ 0, A ∈ BZ

(10.168)

and the corresponding sets of locally uniform convergence for the functions q˜ε,i (·, A), A ∈ Γ given by the following relation: U[q˜ ·,i (· u˜ ·,i, A)] = (0, ∞).

(10.169)

Here, as above, Γ is the class of sets appearing in the condition R. The function q˜0,i (t, A), t ∈ R+, A ∈ BZ obviously belong the class P[Z]. In this case, f0,ii = e0,ii , since p0,ii = 1. Thus, according to the relation (10.89), the distribution function F˜0,i (t) = 1−e−t/e0, ii , t ≥ 0. Therefore, q˜0,i (t, Z) = 1− F˜0,i (t), t ≥ 0, i.e., the function q˜0,i (t, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − F˜0,i (t), t ∈ R+ . In this case, we can apply Theorem 2.3 to the regenerative processes ξε,i (t) with the regenerative lifetimes με,i,+ defined in the relations (10.102)–(10.107). Let us choose an arbitrary 0 ≤ tε → t ∈ (0, ∞) as ε → 0 and let tε = tε u˜ε,i . Obviously, 0 ≤ tε → ∞ as ε → 0 and tε /u˜ε,i = tε → t as ε → 0. Theorem 2.3 applied to the regenerative processes ξε,i (t) with the regenerative lifetimes με,i,+ gives the following relation that holds, for A ∈ Γ and any 0 ≤ tε → t ∈ (0, ∞) as ε → 0: q˜ε,i (tε u˜ε,i, A) = Pε,i,+ (tε u˜ε,i, A) → e−t/e0, ii π0,i (A) as ε → 0.

(10.170)

The relation (10.170) implies that, for s ∈ U[q˜0,i (· u˜ ·,i, A)], A ∈ Γ, us

q˜ε,i (·u˜ε,i, A) −→ q˜0,i (·, A) as ε → 0,

(10.171)

where the functions q˜0,i (·, A), A ∈ Γ, and the sets U[q˜0,i (· u˜ ·,i, A)], A ∈ Γ are given by the relations (10.168) and (10.169). The function q˜0,i (t, A) = e−t/e0, ii π0,i (A), t ≥ 0, A ∈ BZ belongs to the class P[Z] and is consistent with the tail probability function 1 − F˜0,i (t) = e−t/e0, ii , t ≥ 0. ˜ (a) is satisfied. Since, m(U¯ [q˜ ·,i (· u˜ ·,i, A)]) = 0, A ∈ Γ, the relation Thus, condition R ˜ (b) and (c) are satisfied. (10.171) implies that the relations given in the conditions R ˜ Finally, the function q˜0,i (t, A), t ≥ 0 is continuous, for A ∈ Γ and, thus, condition R (d) is also satisfied. The proof of Lemma 10.10 is complete. 

10.2 MARP with removed virtual transitions

221

Remark 10.2 The absence of singular components for the limiting distribution func˜ 1,i, i ∈ X is used in the proof of Lemma 10.10. That is why this tions F0,i j (·), j ∈ Y assumption appears in the condition J1 (b).

10.2.2 Procedure of Partial Removing of Virtual Transitions for Modulating Semi-Markov Processes 10.2.2.1 Multi-Alternating Regenerative Processes with Partially Removed Virtual Transitions for Modulating Semi-Markov Processes. Let us assume that ε ∈ (0, 1] and conditions G1 and H1 hold. Let also K be a domain such that ∅  K ⊆ X. We define the following stopping times for Markov chain ηε,n , for r = 0, 1, . . .:  r +1 if ηε,r  K, θ ε,K [r] = (10.172) min(n > r : ηε,n  ηε,r ) if ηε,r ∈ K. We also define the following successive stopping times: ρε,K,n = θ ε,K [ρε,K,n−1 ], n = 1, 2, . . . , where ρε,K,0 = 0.

(10.173)

Let us now construct a new Markov renewal process (η˜ε,K,n, κ˜ε,K,n ), n = 0, 1, . . . with the phase space X × [0, ∞),  (ηε,0, 0), for n = 0, ρ (η˜ε,K,n, κ˜ε,K,n ) = (η (10.174) , ε,K, n κ ), for n = 1, 2, . . . . ε,ρε,K, n

l=ρε,K, n−1 +1 ε,l

We also define a semi-Markov process, η˜ε,K (t) = η˜ε,K,ν˜ ε,K (t), t ≥ 0,

(10.175)

where, ζ˜ε,K,n = κ˜ε,K,1 + · · · + κ˜ε,K,n, n = 1, 2, . . . , ζ˜ε,K,0 = 0,

(10.176)

are moments of jumps, and ν˜ε,K (t) = max(n ≥ 1 : ζ˜ε,K,n ≤ t),

(10.177)

is the number of jumps in the interval [0, t] for the above semi-Markov process. Let us consider a two component stochastic process, (ξ˜ε,K (t), η˜ε,K (t)) = (ξε (t), η˜ε,K (t)), t ≥ 0.

(10.178)

222

10 Perturbed MARP

This is a multi-alternating regenerative process. It has a regenerative component the same as the original multi-alternating regenerative process (ξε (t), ηε (t)), but has a new modulating semi-Markov process η˜ε,K (t). Moreover, it follows from the definition of the semi-Markov process η˜ε,K (t) that it has trajectories identical to the trajectories of the process ηε (t), i.e., η˜ε,K (t) = ηε (t), t ≥ 0 and, therefore, the new and original multivariate regenerative processes also have identical trajectories, i.e., (ξ˜ε,K (t), η˜ε,K (t)) = (ξε (t), ηε (t)), t ≥ 0.

(10.179)

The difference between these processes lies in their regeneration times ζ˜ε,K,n, n = 0, 1, . . . and ζε,n, n = 0, 1, . . .. A procedure of constructing a family of stochastic triplets satisfying the model assumptions (K)–(O) and such that a multi-alternating regenerative process built using these triplets and the relations (10.1)–(10.3) would have the same finite-dimensional distributions as the multi-alternating regenerative process (ξ˜ε,K (t), η˜ε,K (t)) is similar with those described in Sect. 10.2.1.1. From the definition of the stopping times ρε,K,n it follows that the transition probabilities for the Markov renewal process (η˜ε,K,n, κ˜ε,K,n ) take the following form, for ε ∈ (0, 1]: Q˜ ε,K,i j (t) = P{η˜ε,K,1 = j, κ˜ε,K,1 ≤ t/η˜ε,K,0 = i}  =

¯ j ∈ X, for t ≥ 0, i ∈ K, Q ε,i j (t)  (∗n) ∞ Q˜ ε,i j (t) = n=0 Q ε,ii (t) ∗ Q ε,i j (t) for t ≥ 0, i ∈ K, j ∈ X.

(10.180)

Accordingly, the transition probabilities for the embedded Markov chain η˜ε,K,n are given by the following relation: p˜ε,K,i j = P{η˜ε,K,1 = j/η˜ε,K,0 = i}  =

¯ j ∈ X, for i ∈ K, pε,i j pε, i j p˜ε,i j = I( j  i) p¯ ε, ii for i ∈ K, j ∈ X.

(10.181)

Note that the conditions G1 and H1 guarantee that the probabilities p¯ε,ii > 0, ε ∈ (0, 1], for i ∈ X. We also introduce distribution functions, for i ∈ X,

F˜ε,K,i (t) = (10.182) Q˜ ε,K,i j (t), t ≥ 0. j ∈X

10.2.2.2 Perturbation Conditions for Multi-Alternating Regenerative Processes with Partially Removed Virtual Transitions for Modulating Semi-Markov Processes. Analogues of the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , J◦1 , K1 take for the semi-Markov processes η˜ε,K (t) the forms presented below. These conditions ˜ 1, H ˜ 1 , I˜ 1 , I˜ H,1 , L, ˜ L ˜ H,1 , J˜ 1 , J˜ ◦ , K ˜ 1 for are also direct analogues of the conditions G 1 the semi-Markov processes η˜ε (t).

10.2 MARP with removed virtual transitions

223

Analogues of the conditions G1 and H1 take for the semi-Markov processes η˜ε,K (t) the following forms: ˜  : (a) p˜ε,K,i j > 0, ε ∈ (0, 1] or p˜ε,K,i j = 0, ε ∈ (0, 1], for every i, j ∈ X, (b) G K F˜ε,K,i (0) = 0, for i ∈ X, ε ∈ (0, 1] and ˜  : For any i, j ∈ X, there exist n˜ K,i j ≥ 1 a chain of states i = r˜K,0, r˜K,1, . . . , r˜n˜ K, i j −1 ∈ H K  X, jn˜ K, i j = j such that 1≤l ≤ n˜ K, i j p˜1,K, r˜l−1 r˜l > 0. The following lemma is an analogue of Lemmas 9.31 , 9.51 , and a corollary of Lemma 10.4. Lemma 10.11 The conditions G1 and H1 , assumed to be satisfied for the semiMarkov processes ηε (t), entail the fulfilment of the conditions G1 and H1 for the ˜  and H ˜. semi-Markov processes η˜ε,K (t) in the form of conditions G K K We introduce sets, for i ∈ X and ε ∈ (0, 1], ˜ ε,K,i = { j ∈ X : p˜ε,K,i j > 0}. Y

(10.183)

The following lemma is an analogue of Lemma 9.41 and a corollary of Lemma 10.5. Lemma 10.12 Let the conditions G1 and H1 be satisfied for the semi-Markov pro˜ ε,K,i = Y ˜ 1,K,i, ε ∈ (0, 1], for i ∈ X and, cesses ηε (t). Then, the sets Y  ¯ for i ∈ K, Y1,i ˜ (10.184) Y1,K,i = Y1,i \ {i} for i ∈ K. Analogues of the conditions I1 and IH,1 take for the semi-Markov processes η˜ε,K (t) the following form: I˜ K : p˜ε,K,i j → p˜0,K,i j as ε → 0, for i, j ∈ X and  ˜ 1,K,i, i ∈ X belong to the complete family of I˜ K,H : The functions p˜ ·,K,i j , j ∈ Y asymptotically comparable functions H appearing in the condition IH,1 .

Obviously,  p˜0,K,i j  is a stochastic matrix.  . Note also that the condition I˜ K is implied by the condition I˜ K,H The following lemma is an analogue of Lemma 9.61 . ˜  and I˜  imply that, for i ∈ X, Lemma 10.13 The conditions G K K ˜ 0,K,i = { j ∈ X : p0,K,i j > 0} ⊆ Y ˜ 1,K,i . Y

(10.185)

Let us define a modified set of asymptotically absorbing states, which is an analogue of the set Y0 , (10.186) Y0,K = {i ∈ K : p0,ii = 1}.

224

10 Perturbed MARP

The following lemma is an analogue of Lemma 9.71 and a corollary of Lemma 10.6. Lemma 10.14 The conditions G1 , H1 , I1 , IH,1 , assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of the conditions I1 , IH,1 for  . The limiting the semi-Markov processes η˜ε,K (t) in the form of conditions I˜ K , I˜ K,H  transition probabilities p0,K,i j appearing in the condition I˜ K take the following form:

p˜0,K,i j

⎧ ⎪ p ⎪ ⎨ 0,i j p0, i j ⎪ I( j  i) p¯0, ii = ⎪ ⎪ ⎪ I( j  i) limε→0 ⎩

pε, i j p¯ ε, ii

for i  K, j ∈ X, for i ∈ K \ Y0,K, j ∈ X, for i ∈ Y0,K, j ∈ X.

(10.187)

Note that the condition I˜ K implies that the functions pε,K,i j and pε,K,ii = ki pε,K,ik belong to the family H and, thus, the limits in the relation (10.187) exist. The limiting probabilities p˜0,K,i j appearing in condition I˜ K satisfy the following relations:

p˜0,K,i j ≥ 0, j  i, p˜0,K,ii = 0, i ∈ X, and p˜0,K,i j = 1, i ∈ X. (10.188) 

ji

The normalisation functions u˜ε,K,i, i ∈ X, which are used in the condition J˜ K formulated below take the following forms, for i ∈ X:  ¯ uε,i for i ∈ K, u˜ε,K,i = −1 (10.189) p¯ε,ii uε,i for i ∈ K. ˜  and H ˜  imply that p¯ε,K,ii ∈ (0, 1], ε ∈ 0, 1], for i ∈ X. Note that the conditions G K K Thus, by virtue of the condition J1 (c), for i ∈ X, u˜ε,K,i ∈ (0, ∞), ε ∈ (0, 1].

(10.190)

Analogue of the conditions L1 and LH,1 take for the semi-Markov processes η˜ε,K (t) the following forms: ˜ K : u˜ε,K,i → u˜0,K,i ∈ (0, ∞] as ε → 0, for i ∈ X L and  ˜ H,K L : Functions u˜ ·,K,i, i ∈ X belong to the complete family of asymptotically comparable functions H appearing in condition LH,1 .

The following lemma is an analogue of Lemma 9.121 and a corollary of Lemma 10.7. Lemma 10.15 The conditions G1 , H1 and I1 , IH,1 , L1 , LH,1 , assumed to be satisfied for the semi-Markov process ηε (t), entail the fulfilment of the conditions L1 and LH,1  ˜ K and L ˜ H,K for the semi-Markov processes η˜ε (t) in the form of conditions L .

10.2 MARP with removed virtual transitions

225

The distribution function F˜ε,K,i j (t), t ≥ 0, which is an analogue of the distribution ˜ 1,K,i, i ∈ X and function Fε,i j (t), t ≥ 0, is defined by the following relation, for j ∈ Y ε ∈ (0, 1]: F˜ε,K,i j (t) = P{ κ˜ε,K,1 ≤ t/η˜ε,K,0 = i, η˜ε,K,1 = j} ⎧ Fε,i j (t) ⎪ ⎪ ⎪ ˜ 1,K,i, i ∈ K, ¯ ⎨ ⎪ for t ≥ 0, j ∈ Y = ˜ (∗n) 1 ∞ n p Fε,i j (t) = p˜ ε, i j n=0 Fε,ii (t) ∗ Fε,i j (t)pε,ii ⎪ ε,i j ⎪ ⎪ ⎪ ˜ for t ≥ 0, j ∈ Y1,K,i, i ∈ K. ⎩

(10.191)

The corresponding Laplace transform φ˜ε,K,i j (s) takes the following form, for ˜ 1,K,i, i ∈ X and ε ∈ (0, 1]: j∈Y ∫ ∞ e−st F˜ε,K,i j (dt) φ˜ε,K,i j (s) = E{e−s κ˜ε,K,1 /η˜ε,K,0 = i, η˜ε,K,1 = j} = 0

=

φε,i j (s) φ˜ε,i j (s) =

φ ε, i j (s) p¯ ε, ii 1−φ ε, ii (s)pε, ii

˜ 1,K,i, i ∈ K, ¯ for s ≥ 0, j ∈ Y ˜ 1,K,i, i ∈ K. for s ≥ 0, j ∈ Y (10.192)

Also, we can define the corresponding distribution function F˜ε,K,i j (u) and its ˜ 1,K,i, i ∈ X, Laplace transform φ˜ε,K,i j (s), for ε ∈ (0, 1] and j  Y F˜ε,K,i j (t) = F˜ε,K,i (t), t ≥ 0 ∫

and φ˜ε,K,i j (s) = φ˜ε,K,i (s) =

0

∞

e−su F˜ε,K,i (du), s ≥ 0.

(10.193)

(10.194)

¯ 1,K,i, i ∈ X does not affect the Note that the above choice of F˜ε,K,i j (t) for j ∈ Y ˜ transition probabilities Q ε,K,i j (t) of the semi-Markov process η˜ε,K (t) and analogues of the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , J◦1 , and K1 for the semi-Markov processes η˜ε,K (t) (see, Sect. 10.1.1.1 and Remark 10.1). Analogues of the conditions J1 and J◦1 take for the semi-Markov processes η˜ε,K (t) the following forms: ˜ 1,K,i, i ∈ X, (b) F˜0,K;i j (·) J˜ K : (a) F˜ε,K,i j (· u˜ε,K,i ) ⇒ F˜0,K,i j (·) as ε → 0, for j ∈ Y is a non-arithmetic distribution function without singular component, for j ∈ ˜ 1,K,i, i ∈ X, (c) u˜ε,K,i ∈ (0, ∞), ε ∈ (0, 1], for i ∈ X Y and ˜ ˜ J˜ K◦ : (a) φ˜ε,K,i j (s/ ∫ ∞u˜ε,K,i ) → φ0,K,i j (s) as ε → 0, for s ≥ 0 and j ∈ Y1,K,i, i ∈ X, (b) −st ˜ ˜ φ0,K,i j (s) = 0 e F0,K,i j (dt), s ≥ 0 is the Laplace transform of a non-arithmetic ˜ 1,K,i, i ∈ X, distribution function without singular component F˜0,K,i j (·), for j ∈ Y (c) u˜ε,K,i ∈ (0, ∞), ε ∈ (0, 1], for i ∈ X.

226

10 Perturbed MARP

The following lemma is an analogue of Lemma 9.101 and a corollary of Lemma 10.7. Lemma 10.16 The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J◦1 , (J1 ), K1 , assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of the condition J◦1 (J1 ) for the semi-Markov processes η˜ε,K (t) in the form of condition J˜ K◦ (J˜ K ), where: ˜ 1,K,i, i ∈ X appearing in the (i) The limiting Laplace transforms φ˜0,K,i j (·), j ∈ Y condition J˜ K◦ take the following forms: φ˜0,K,i j (s) ⎧ φ0,i j (s) ⎪ ⎪ ⎨ ⎪ = φ˜0,i j (s) = ⎪ ⎪ ⎪ φ˜0,i j (s) = ⎩

φ0, i j (s p¯ 0, ii ) p¯ 0, ii 1−φ0, ii (s p¯ 0, ii )p0, ii 1 1+ f0, ii s

˜ 1,K,i, i ∈ K, ¯ for s ≥ 0, j ∈ Y ˜ 1,K,i, i ∈ K \ Y0,K, for s ≥ 0, j ∈ Y ˜ 1,K,i, i ∈ Y0,K . for s ≥ 0, j ∈ Y

(10.195)

(ii) The normalisation functions u˜ ·,K,i, i ∈ X appearing in the condition JK◦ are given by the relation (10.189). The expectations f˜ε,K,i j , which are analogues of the expectations fε,i j are defined, ˜ 1,K,i, i ∈ X: for ε ∈ (0, 1] by the following relation, for j ∈ Y ∫ ∞ u F˜ε,K,i j (du) f˜ε,K,i j = 0

 =

˜ 1,K,i, i ∈ K, ¯ fε,i j for j ∈ Y −1 f ˜ f˜ε,i j = fε,i j + pε,ii p¯ε,ii for j ∈ Y , i ∈ K. ε,ii 1,K,i

(10.196)

An analogue of the condition K1 takes for the semi-Markov processes η˜ε,K (t) the following form: ˜ 1,K,i, i ∈ X, for every ε ∈ (0, 1], (b) f˜ε,K,i j /u˜ε,K,i → f˜0,K,i j ˜  :(a) f˜ε,K,i j < ∞, j ∈ Y K K ∫∞ ˜ 1,K,i, i ∈ X. = 0 u F˜0,K,i j (du) < ∞ as ε → 0, for j ∈ Y The following lemma is an analogue of Lemma 9.111 and a corollary of Lemma 10.8. Lemma 10.17 The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of the condition ˜  , where: K1 for the semi-Markov processes η˜ε,K (t) in the form of condition K K ˜ 1,K,i, i ∈ X appearing in the condition (i) The limiting expectations f˜0,K,i j , j ∈ Y ˜  are given by the following relation: K K 

f˜0,K,i j =

˜ 1,K,i, i ∈ K, ¯ f0,i j for j ∈ Y ˜f0,i j = p¯0,ii f0,i j + p0,ii f0,ii for j ∈ Y ˜ 1,K,i, i ∈ K.

(10.197)

˜  are (ii) The normalisation functions u˜ ·,K,i, i ∈ X appearing in the condition K K given by the relation (10.189).

10.2 MARP with removed virtual transitions

227

The functions q˜ε,K,i (t, A), which are analogue of functions q˜ε,i (t, A) are defined, for ε ∈ (0, 1] by the following relation, for i ∈ X: q˜ε,K,i (t, A) = Pi { ξ˜ε,K (t) ∈ A, ζ˜ε,K,1 > t}  ¯ A ∈ BZ, q (t, A) for t ≥ 0, i ∈ K, = ε,i q˜ε,i (t, A) for t ≥ 0, i ∈ K, A ∈ BZ .

(10.198)

The relation (10.198) implies that functions q˜ε,K,i (t u˜ε,K,i, A), t ∈ R+, A ∈ BZ belong to the class P[BZ ], for i ∈ X, ε ∈ (0, 1]. Moreover, the function q˜ε,K,i (t u˜ε,K,i, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − F˜ε,K,i (t u˜ε,K,i ), for i ∈ X and ε ∈ (0, 1], i.e., q˜ε,K,i (t u˜ε,K,i, Z) = 1 − F˜ε,K,i (t u˜ε,K,i ), for t ∈ R+ .

(10.199)

An analogue of the condition R takes for the multi-alternating regenerative processes (ξ˜ε,K (t), η˜ε,K (t)) the following form: ˜  : There exist functions q˜0,K,i (t, A), t ≥ 0, A ∈ BZ , i ∈ X, which belong to the R K class P[BZ ], a class of set Γ ⊆ BZ , and Borel sets U[q˜ ·,K,i (· u˜ ·,K,i, A)], A ∈ Γ, i ∈ X such that: (a) The function q˜0,K,i (t, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − F˜0,K,i (t), t ∈ R+ , for i ∈ X; (b) the functions us q˜ε,K,i (· u˜ε,K,i, A) −→ q˜0,K,i (·, A) as ε → 0, for points s ∈ U[q˜ ·,K,i (· u˜ ·,K,i, A)], A ∈ Γ, i ∈ X; (c) m(U¯ [q˜ ·,K,i (· u˜ ·,K,i, A)]) = 0, for A ∈ Γ, i ∈ X; (d) the function q˜0,K,i (·, A) is continuous almost everywhere with respect to the Lebesgue measure m(·) on B+ , for A ∈ Γ, i ∈ X. ¯ 1,K,i, i ∈ X Note also that the choice of the distribution functions F˜ε,K,i j (·), j  Y  ˜ according to the relation (10.193) does not affect the condition RK since the functions q˜ε,K,i (·, A), i ∈ X, A ∈ BZ do not depend on the above distribution functions. The following lemma is a corollary of Lemma 10.10. Lemma 10.18 The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , and R, assumed to be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)), entail the fulfilment of the condition R for the multi-alternating regenerative processes ˜  , where: (ξ˜ε,K (t), η˜ε,K (t)) in the form of condition R K

˜  coincides with (i) The limiting function q˜0,K,i (·, A) appearing in the condition R K ¯ or is defined the function q0,i (·, A) appearing in the condition R, for A ∈ BZ, i ∈ K, for A ∈ BZ, i ∈ K by the relation (10.126), if p¯0,ii = 1, or by the relation (10.135), if p¯0,ii ∈ (0, 1), or by the relation (10.168), if p¯0,ii = 0, for A ∈ BZ, i ∈ K. ˜  coincides with the class of (ii) The class of sets Γ appearing in the condition R K sets Γ appearing in the condition R. (iii) The set of locally uniform convergence U[q˜ ·,K,i (· u˜ ·,K,i, A)] appearing in the ˜  coincides with the set U[q ·,i (· u ·,i, A)] appearing in condition R, for condition R K ¯ or is defined A ∈ Γ, i ∈ K by the relation (10.127), if p¯0,ii = 1, or by A ∈ Γ, i ∈ K, the relation (10.158), if p¯0,ii ∈ (0, 1), or by the relation (10.169), if p¯0,ii = 0.

228

10 Perturbed MARP

˜  are (iv) The normalisation functions u˜ ·,K,i, i ∈ X appearing in the condition R K given by the relation (10.189).

10.3 Multi-Alternating Regenerative Processes with Reduced Modulating Semi-Markov Processes In this section, asymptotic procedures of aggregation of regeneration times for perturbed multi-alternating regenerative processes, based on one-state reduction of phase space for modulating semi-Markov processes with totally or partially removed virtual transitions, are described.

10.3.1 Procedure of One-State Reduction of Phase Space for Modulating Semi-Markov Processes 10.3.1.1 Multi-Alternating Regenerative Processes with Reduced Phase Space for Modulating Semi-Markov Processes. Let us assume that ε ∈ (0, 1] and the conditions G1 and H1 are satisfied. Let us choose some state k ∈ X and introduce the reduced phase space k X = X \ {k}. We define the following stopping times for the Markov chain η˜ε,n , for r = 0, 1, . . .: k αε [r]

= min(n > r : η˜ε,n ∈ k X).

(10.200)

By the definition, k αε [r] is the first after r time of hitting in the reduced phase space k X by the Markov chain η˜ε,n . Since the Markov chain η˜ε,n does not make virtual transitions (of the form i → i for i ∈ X), the following relation takes place, for r = 0, 1, . . .,  r + 1 if η˜ε,r+1 ∈ k X, (10.201) k αε [r] = r + 2 if η˜ε,r+1 = k. Let us also define the following sequential stopping times for Markov chains η˜ε,n k βε,n

= k αε [k βε,n−1 ], n = 1, 2, . . . , where k βε,0 = I(η˜ε,0 = k).

(10.202)

Let us now construct a new Markov renewal process (k ηε,n, k κε,n ), n = 0, 1, . . ., with the phase space k X × [0, ∞), (η˜ε, k βε,0 , 0) for n = 0, k βε, n (k ηε,n, k κε,n ) = (10.203) (η˜ε, k βε, n , l= k βε, n−1 +1 κ˜ε,l ) for n = 1, 2, . . . . and define the corresponding reduced modulating semi-Markov process,

10.3 MARP with reduced modulating SMP k ηε (t)

229

= k ηε, k νε (t), t ≥ 0,

(10.204)

where, k ζε,n

= k κε,1 + · · · + k κε,n, n = 1, 2, . . . , k ζε,0 = 0,

(10.205)

are the corresponding instants of jumps, and, k νε (t)

= max(n ≥ 1 : k ζε,n ≤ t),

(10.206)

is the number of jumps in the interval [0, t] for the above semi-Markov process. By the definition of the semi-Markov process k ηε (t), the following relation holds, for i ∈ X, Pi {k ηε (t) ∈ k X, t ≥ 0} = 1. (10.207) The relation (10.207) allows us to consider k ηε (t) as a semi-Markov process with a reduced phase space k X and refer to k ηε (t) as a reduced modulating semi-Markov process. Let us consider a two component stochastic process, (k ξε (t), k ηε (t)) = (ξε (ζ˜ε, k βε,0 + t), k ηε (t)), t ≥ 0.

(10.208)

This is a multi-alternating regenerative process. It has a regenerative component the same as the original multi-alternating regenerative process (ξε (t), ηε (t)), but has a new modulating semi-Markov process k ηε (t). Let us also describe a procedure of constructing a family of stochastic triplets satisfying the model assumptions (K)–(O) and such that a multi-alternating regenerative process built using these triplets and the relations (10.1)–(10.3) would have the same finite-dimensional distributions as the multi-alternating regenerative process (k ξε (t), k ηε (t)).  = ξ  (t), t ≥ 0 , κ  , η  stochastic triplet ξ¯ = Let us denote by ξ¯ε,i ε ε,i ε,i ε,i k ξε (t), t ≥ 0 , k κε,1, k ηε,1 , for the case, where the initial state k ηε (0) = i, for i ∈ k X.   (t), t ≥ 0 , κ  , η  , = ξε,i,n Let also a family of stochastic triplets ξ¯ε,i,n ε,i,n ε,i,n  i ∈ k X, n = 1, 2, . . . and a random variable ηε satisfying assumptions (K)–(O) are  = ξ  (t), t ≥ 0 , κ  , η  , i ∈ constructed as probability copies of the triplet ξ¯ε,i ε,i ε,i ε,i k X and the random variable ηε . The multi-alternating regenerative process (ξε(t), ηε(t)), t ≥ 0, built using the     = ξε,i,n (t), t ≥ 0 , κε,i,n , ηε,i,n

, i ∈ k X, n = 1, 2, . . ., the above triplets ξ¯ε,i,n  random variable ηε , and the relations (10.1)–(10.3), is a probabilistic copy of the process (k ξε (t), k ηε (t)), t ≥ 0. This means that these processes have the same finitedimensional distributions, i.e., (k ξε (t), k ηε (t)), t ≥ 0 = (ξε(t), ηε(t)), t ≥ 0. d

(10.209)

The above relation holds for any distribution of the random variable ηε = ηε (0), including the case, where the following condition holds: k W:

P{ηε = ηε (0) ∈ k X} = 1, for ε ∈ (0, 1].

230

10 Perturbed MARP

In this case, P{k βε,0 = 0} = P{ ζ˜ε, k βε,0 = 0} = 1, and, thus, P{k ηε (0) = ηε (0)} = 1, and the following relation takes place for the multi-alternating regenerative process (k ξε (t), k ηε (t)), for i ∈ k X: Pi {(k ξε (t), k ηε (t)) = (ξ˜ε (t), k ηε (t)) = (ξε (t), k ηε (t)), t ≥ 0} = 1.

(10.210)

According to the relations (10.201)–(10.206) and (10.210), the multi-alternating regenerative process (k ξε (t), k ηε (t)) can be viewed as the result of a two-stage procedure applied to the initial multi-alternating regenerative process (ξε (t), ηε (t)). At the first stage, the multi-alternating regenerative process (ξε (t), ηε (t)) is transformed into the multi-alternating regenerative process (ξ˜ε (t), η˜ε (t)), using the procedure of total removing of virtual transitions from trajectories of the modulating semi-Markov process ηε (t). At the second stage, the multi-alternating regenerative process (ξ˜ε (t), η˜ε (t)) is transformed into the multi-alternating regenerative process(k ξε (t), k ηε (t)), using the above phase space reduction procedure applied to the modulating semi-Markov process η˜ε (t). 10.3.1.2 Perturbation Conditions for Reduced Modulating Semi-Markov Process. In this and the next subsections, we formulate the conditions for the perturbed modulating semi-Markov process k η(t). For simplicity, we assume that the condition k W is satisfied. The transition probabilities for the Markov renewal process (k ηε,n, k κε,n ) take the following form, for t ≥ 0, i, j ∈ k X: k Q ε,i j (t)

= P{k ηε,1 = j, k κε,1 ≤ t/ k ηε,0 = i} = Q˜ ε,i j (t) + Q˜ ε,ik (t) ∗ Q˜ ε,k j (t).

(10.211)

The transition probabilities for the embedded Markov chain k ηε,n are given by the following relation, for i, j ∈ k X: k pε,i j

= P{k ηε,1 = j/ k ηε,0 = i} = p˜ε,i j + p˜ε,ik p˜ε,k j .

(10.212)

Let us also introduce the distribution functions of sojourn times for semi-Markov process k ηε (t), for i ∈ k X,

(10.213) k Q ε,i j (t), t ≥ 0. k Fε,i (t) = j∈ kX

In Sect. 9.31 , we have described transformation of the conditions G, H, I, IH , L, LH , J, J◦ , and K for semi-Markov processes ηε (t) in their analogues k G, k H, k I, k IH , k L, k LH , k J, k J, and k K for the semi-Markov processes k ηε (t). Similarly, the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , J◦1 , and K1 can be transformed in their analogues k G1 , k H1 , k I1 , k IH,1 , k L1 , k LH,1 , k J1 , k J◦1 , and k K1 . Analogues of the conditions G1 and H1 take for the semi-Markov processes k ηε (t) the following forms:

10.3 MARP with reduced modulating SMP k G1 :

(a) k pε,i j > 0, ε ∈ (0, 1] or k pε,i j = 0, ε ∈ (0, 1], for every i, j ∈ = 0, for i ∈ k X, ε ∈ (0, 1]

k Fε,i (0)

231 k X,

(b)

and k H1 :

For any i, j ∈

k r0, k r1, . . . , k rk ni j

a chain of states i = k X, there exist k ni j ≥ 1 and  = j, all from space k X, such that 1≤l ≤ k ni j k p1, k rl−1 k rl > 0.

The following lemma is a direct corollary of Lemma 9.141 . Lemma 10.19 The conditions G1 and H1 , assumed to be satisfied for the semiMarkov processes ηε (t), entail the fulfilment of the conditions G1 and H1 for the semi-Markov processes k ηε (t) in the forms of the conditions k G1 and k H1 . The following sets were introduced in the relations (9.81)1 and (9.82)1 : k Y1,i

= { j ∈ k X : k p1,i j > 0}  ˜ 1,i ∪ Y ˜ 1,k ) \ {k} if k ∈ Y ˜ 1,i, i ∈ k X, (Y = ˜ 1,i ˜ 1,i, i ∈ k X. Y if k  Y

(10.214)

Analogues of the conditions I1 and IH,1 take for the semi-Markov processes the following forms:

k ηε (t)

k I1 : k pε,i j

= p˜ε,i j + p˜ε,ik p˜ε,k j → k p0,i j = p˜0,i j + p˜0,ik p˜0,k j as ε → 0, for i, j ∈ k X

and The functions k p ·,i j , j ∈ k Y1,i, i ∈ k X belong to the complete family of asymptotically comparable functions H appearing in condition IH,1 .

k IH,1 :

Obviously, k p0,i j  is a stochastic matrix. Note also that the condition k I1 is implied by the condition k IH,1 . The following lemma is a direct corollary of Lemma 9.171 . Lemma 10.20 The conditions G1 , H1 , I1 , and IH,1 , assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of the conditions I1 and IH,1 for the semi-Markov processes k ηε (t) in the form of conditions k I1 and k IH,1 . The normalisation functions k uε,i, i ∈ X, which used in the condition k J take the following form, for i ∈ k X: k uε,i

−1 = u˜ε,i = p¯−1 ε,ii uε,i = (1 − pε,ii ) uε,i .

(10.215)

Note that from the conditions G1 and H1 it follows that p¯ε,ii ∈ (0, 1], ε ∈ 0, 1], for i ∈ X and, thus, due to the above conditions and the condition J1 (c), for i ∈ k X, k uε,i

∈ (0, ∞), ε ∈ (0, 1].

(10.216)

Analogues of the conditions L1 and LH,1 take for the semi-Markov processes η k ε (t) the following forms: k L1 : k uε,i

→ k u0,i ∈ (0, ∞] as ε → 0, for i ∈ k X

232

10 Perturbed MARP

and functions k u ·,i, i ∈ k X belong to the complete family of asymptotically comparable functions H appearing in the condition LH,1 .

k LH,1 : The

The following lemma is a direct corollary of Lemma 10.6. Lemma 10.21 The conditions G1 , H1 and I1 , IH,1 , L1 , LH,1 , assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of the conditions L1 and LH,1 for the semi-Markov processes k ηε (t) in the form of conditions k L1 and k LH,1 . Note, also, that the conditions G1 , H1 , I1 , L1 , and the relation (10.78) imply that the limits k u0,i, i ∈ k X appearing in the condition k L1 take the following forms:  −1 p¯0,ii u0,i ∈ (0, ∞) if p¯0,ii > 0, u0,i ∈ (0, ∞), (10.217) u = k 0,i ∞ if p¯0,ii = 0 or p¯0,ii > 0, u0,i = ∞. The corresponding distribution functions k Fε,i j (u) and their Laplace transform φ k ε,i j (s) given, respectively, by relations (9.83)1 and (9.86)1 , take the following forms, for ε ∈ (0, 1] and j ∈ k Y1,i, i ∈ k X: p˜ε,i j p˜ε,ik p˜ε,k j + F˜ε,ik (t) ∗ F˜ε,k j (t) ,t ≥ 0 k pε,i j k pε,i j

(10.218)

p˜ε,i j p˜ε,ik p˜ε,k j + φ˜ε,ik (s)φ˜ε,k j (s) , s ≥ 0. p k ε,i j k pε,i j

(10.219)

k Fε,i j (t)

= F˜ε,i j (t)

k φε,i j (s)

= φ˜ε,i j (s)

and

Also, we can define the corresponding distribution function k Fε,i j (u) and its ¯ 1,i, i ∈ k X, Laplace transform k φε,i j (s), for ε ∈ (0, 1] and j ∈ k Y k Fε,i j (t)

∫

and k φε,i j (s) = k φε,i (s) =

0

= k Fε,i (t), t ≥ 0 ∞

e−su k Fε,i (du), s ≥ 0.

(10.220)

(10.221)

¯ 1,i, i ∈ k X does not affect the transition Note that the choice of k Fε,i j (·), j ∈ k Y probabilities k Q ε,i j (t) of the semi-Markov process k ηε (t) and analogues of the conditions k G1 , k H1 , k I1 , k IH,1 , k L1 , k LH,1 , J1 , and k K1 for the semi-Markov processes k ηε (t) (see, Sect. 10.1.1.1 and Remark 10.1). Analogues of the conditions J1 and J◦1 take for the semi-Markov processes k ηε (t) the following forms: k J1 :(a) k Fε,i j (· k uε,i )

⇒ k F0,i j (·) as ε → 0, for j ∈ k Y1,i, i ∈ k X, (b) k F0,i j (·) is a non-arithmetic distribution function without singular component, for j ∈ k Y1,i , i ∈ k X, (c) k uε,i ∈ (0, ∞), ε ∈ (0, 1], for i ∈ k X

and

10.3 MARP with reduced modulating SMP ◦ k J1 :

233

(a) k φε,i j (s/ k uε,i ) → k φ0,i j (s) as ε → 0, for s ≥ 0 and j ∈ k Y1,i, i ∈ k X, (b) k φ0,i j (·) is the Laplace transform of non-arithmetic distribution function without singular component k F0,i j (·), for j ∈ k Y1,i, i ∈ k X, (c) k uε,i ∈ (0, ∞), ε ∈ (0, 1], for i ∈ k X.

The conditions k G1 , k H1 , and k I1 imply that, for i ∈ k X, k Y0,i

⊆ k Y1,i .

(10.222)

Below, we shall show that the normalisation functions k uε,i, i ∈ k X defined by the relation (10.215) can be used in the conditions k J1 and k J◦1 . The conditions k G1 , k H1 , k I1 , k J1 , and the relation (10.222) imply that, for i ∈ k X,



k Fε,i (·k uε,i ) = k Q ε,i j (· k uε,i ) = k Fε,i j (· k uε,i ) k pε,i j j ∈ k Y1, i



⇒

j ∈ k Y1, i k F0,i j (·) k p0,i j

j ∈ k Y1, i

=



k F0,i j (·) k p0,i j

j ∈ k Y0, i

= k F0,i (·) as ε → 0.

(10.223)

The conditions k G1 , k H1 , k I1 , k J1 and the relation (10.222) imply that, for i, j ∈ k X, k Q ε,i j (·)

= k Fε,i j (·) k pε,i j ⇒ k F0,i j (·) k p0,i j = k Q0,i j (·) as ε → 0.

Note that k Q 0,i j (·)

 =

k F0,i j (·) k p0,i j for j ∈ k Y0,i, i ∈ k X, ¯ 0,i, i ∈ k X. 0 for j ∈ k Y

(10.224)

(10.225)

Lemma 9.201 implies that, under the condition k IH,1 , the following asymptotic ˆ formulated in Sect. 9.3.2.21 ) holds, for i, j ∈ X: relation (given in the condition L wε, j,i =

u˜ε, j → w0, j,i ∈ [0, ∞] as ε → 0. u˜ε,i

(10.226)

Let us introduce the following set: W0 = { j ∈ X : w0, j,i ∈ [0, ∞), i ∈ X}.

(10.227)

W0  ∅.

(10.228)

Obviously, This relation lets us choose the least absorbing state k, for which the condition (formulated in Sect. 9.3.2.21 ) holds. This condition can be re-formulated in the following equivalent form:

kM

234 k M:

10 Perturbed MARP

k ∈ W0 , i.e, limε→0

u˜ ε, k u˜ ε, i

= w0,k,i ∈ [0, ∞), for i ∈ X.

The following lemma is an analogue of Lemma 9.211 . Lemma 10.22 The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J◦1 (J1 ), K1 , and k M, assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of the condition J◦1 (J1 ) for the semi-Markov processes k ηε (t) in the form of condition ◦ k J1 (k J1 ), where: (i) The limiting Laplace transforms k φ0,i j (·), j ∈ k Y1,i, i ∈ k X appearing in the condition k J◦1 are given by the relation (10.230). (ii) The normalisation functions k u ·,i, i ∈ k X appearing in the condition k J◦1 (k J1 ) are given by the relation (10.215). Proof Lemma 9.191 implies that, under the condition IH,1 , the following asymptotic relation (given in condition k Iˆ formulated in Sect. 9.3.2.21 ) holds, for i, j ∈ k X: k qε,i j

=

p˜ε,i j → k q0,i j ∈ [0, 1] as ε → 0. k pε,i j

(10.229)

Lemma 9.211 entails the fulfilment of the asymptotic relation appearing in the condition k J◦1 . Wherein, the corresponding limiting distribution functions k F0,i j (·), given by the relation (9.91)1 , take the following forms, for j ∈ k Y1,i, i ∈ k X: k φε,i j (s/ k uε,i )

→ k φ0,i j (s) = φ˜0,i j (s) k q0,i j + φ˜0,ik (s)φ˜0,k j (w0,k,i s)(1 − k q0,i j ) as ε → 0, for s ≥ 0. (10.230)

Note that, according to Lemma 9.211 , the function k φ0,i j (s) is the Laplace transform of the limiting distribution function k F0,i j (·) such that k F0,i j (0) < 1, for j ∈ k Y1,i, i ∈ k X. Obviously, the relation (10.230) appearing in the condition k J◦1 and proved in Lemma 9.211 can be re-written in the equivalent form using characteristic functions instead of Laplace transform. The corresponding limiting characteristic function k ϕ0,i j (z) of the limiting distribution function k F0,i j (·) takes, in this case, the following form, for j ∈ k Y1,i, i ∈ k X: k φ0,i j (z)

= ϕ˜0,i j (z) k q0,i j + ϕ˜0,ik (z)ϕ˜0,k j (w0,k,i z)(1 − k q0,i j ), for z ∈ R1 .

(10.231)

˜ 1,i and, thus, according to the condition J˜ 1 the If k q0,i j ∈ (0, 1], then j ∈ Y ˜ distribution function F0,i j (·) is non-arithmetic. By Lemma A.1, this implies that | ϕ˜0,i j (z)| < 1, z  0 and, thus, |k ϕ0,i j (z)| ≤ | ϕ˜0,i j (z)| k q0,i j + | ϕ˜0,ik (z)|| ϕ˜0,k j (w0,k,i z)|(1 − k q0,i j ) < k q0,i j + 1 − k q0,i j = 1, z ∈ R1 .

(10.232)

10.3 MARP with reduced modulating SMP

235

˜ 1,i, j ∈ Y ˜ 1,k and, thus, according to the condition J˜ 1 , the If k q0,i j = 0, then k ∈ Y ˜ distribution functions F0,ik (·) and F˜0,ik (·) are non-arithmetic. By Lemma A.1, this implies that | ϕ˜0,ik (z)|, | ϕ˜0,k j (z)| < 1, z  0 and, thus, |k ϕ0,i j (z)| ≤ | ϕ˜0,ik (z)|| ϕ˜0,k j (w0,k,i z)| < 1, z  0.

(10.233)

The relations (10.231), (10.232) and (10.233) imply, by Lemma A.1, that the distribution function k F0,i j (·) is non-arithmetic, for j ∈ k Y1,i, i ∈ X. The relation (10.230) imply that the distribution function k F0,i j (·) has the following form, for j ∈ k Y1,i, i ∈ X: k F0,i j (·)

−1 = F˜0,i j (·) k q0,i j + F˜0,ik (·) ∗ F˜0,k j (w0,k,i ·)(1 − k q0,i j ).

(10.234)

˜ 1,i , and, thus, according to the condition J˜ 1 , the distribution If k q0,i j = 1, then j ∈ Y function k F0,i j (·) = F˜0,i j (·) has no singular component. ˜ 1,i, j ∈ Y ˜ 1,k , and, thus, according to the condition J˜ 1 , If k q0,i j = 0, then k ∈ Y the distribution functions F˜0,ik (·), F˜0,ik (·) and, therefore, the distribution function k F0,i j (·) = F˜0,ik (·) ∗ F˜0,ik (·) have no singular component. ˜ 1,i, k ∈ Y ˜ 1,i, j ∈ Y ˜ 1,k and, thus, according to the If k q0,i j ∈ (0, 1) then j ∈ Y condition J˜ 1 , the distribution functions F˜0,i j (·), F˜0,ik (·), F˜0,ik (·), and, therefore, the distribution function k F0,i j (·) given by relation (10.234) have no singular component. The proof is complete.  The corresponding expectation f˜ε,i j are given by relation (9.65)1 and take the following form, for ε ∈ (0, 1] and j ∈ k Y1,i, i ∈ k X: ∫ ∞ u k Fε,i j (du) k fε,i j = 0

= fε,i j pε,i j + ( fε,ik + fε,k j )pε,ik pε,k j .

(10.235)

An analogue of the condition K1 takes for the semi-Markov processes k ηε (t) the following form: (a) k ∫fε,i j < ∞, j ∈ k Y1,i, i ∈ X, for every ε ∈ (0, 1], (b) ∞ f k 0,i j = 0 u k F0,i j (du) < ∞ as ε → 0, for j ∈ k Y1,i, i ∈ k X.

k K1 :

k fε,i j / k uε,i

→

The following lemma is a direct corollary of Lemma 9.221 . Lemma 10.23 The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , and k M, assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of the condition K1 for the semi-Markov processes k ηε (t) in the form of condition k K1 , where: (i) The limiting expectations k f0,i j , j ∈ k Y1,i, i ∈ k X appearing in the condition K k 1 are given by the relation (10.236). (ii) The normalisation functions k u ·,i, i ∈ k X.appearing in the condition k K1 are given by the relation (10.215).

236

10 Perturbed MARP

Proof Lemma 9.221 implies holding of the asymptotic relation appearing in the condition k K1 , with the limits k f0,i j given by the relation (9,97)1 , for j ∈ k Y1,i, i ∈ k X. This relation takes the following form, for j ∈ k Y1,i, i ∈ k X: k fε,i j k uε,i

f˜ε,ik f˜ε,i j p˜ε,i j f˜ε,k j u˜ε,k p˜ε,ik p˜ε,k j + + u˜ε,i k pε,i j u˜ε,i u˜ε,k u˜ε,i k pε,i j ˜ ˜ ˜ → k f0,i j = f0,i j k q0,i j + ( f0,ik + f0,k j w0,k,i )(1 − k q0,i j ) ∫ ∞ = u k F0,i j (du) as ε → 0. (10.236) =

0



The proof is complete.

10.3.1.3 The Perturbation Condition R for Multi-Alternating Regenerative Processes with Reduced Modulating Semi-Markov Process. The form of the functions qε,i (tuε,i, A) used in the condition R suggests that the role of these functions for the multi-alternating regenerative processes (k ξε (t), k ηε (t)) can execute functions, k qε,i (t k uε,i ,

A), t ≥ 0, i ∈ k X, A ∈ BZ,

where the normalisation functions k uε,i, i ∈ and, k qε,i (t,

kX

(10.237)

are given by the relation (10.215)

A) = Pi {k ξε (t) ∈ A, k ζε,1 > t}, t ≥ 0, i ∈ k X, A ∈ BZ .

(10.238)

The relation (10.238) implies that the functions k qε,i (t k uε,i, A), t ∈ R+, A ∈ BZ belong to the class P[BZ ], for i ∈ k X, ε ∈ (0, 1]. Moreover, the function k qε,i (t, k uε,i, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − k Fε,i (t k uε,i ), for i ∈ k X and ε ∈ (0, 1], i.e., k qε,i (t k uε,i, Z)

= 1 − k Fε,i (t k uε,i ), for t ∈ R+ .

(10.239)

An analogue of the condition R takes for the multi-alternating regenerative processes (k ξε (t), k ηε (t)) the following form: There exist functions k q0,i (t, A), t ≥ 0, A ∈ BZ , i ∈ k X, which belong to the class P[BZ ], a class of set Γ ⊆ BZ , and Borel sets U[k q ·,i (· k u ·,i, A)], A ∈ Γ, i ∈ k X such that: (a) the function k q0,i (t, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − k F0,i (t), t ∈ R+ , for i ∈ k X; (b) the functions us k qε,i (· k uε,i, A) −→ k q0,i (·, A) as ε → 0, for points s ∈ U[k q ·,i (· k u ·,i, A)], A ∈ Γ, i ∈ k X; (c) m(U¯ [k q ·,i (· k u ·,i, A)]) = 0, for A ∈ Γ, i ∈ k X; (d) the function k q0,i (·, A) is continuous almost everywhere with respect to the Lebesgue measure m(·) on B+ , for A ∈ Γ, i ∈ k X.

k R:

It should be noted that the notation U[k q ·,i (· k u ·,i, A)] is used to indicate that this set of convergence is determined by the family of functions k qε,i (· k uε,i, A), ε ∈ (0, 1]. It is also helpful to make a few comments about the above perturbation condition. In light of the relation (10.239), the consistency condition k R (a) is a natural.

10.3 MARP with reduced modulating SMP

237

The consistency relation (10.239) implies that, for every ε ∈ (0, 1] and A ∈ BZ, i ∈ A) is majorized by the tail probability function 1 − k Fε,i (· k uε,i ) on interval [0, ∞), i.e., k X, function k qε,i (·,

k qε,i (t k uε,i,

A) ≤ 1 − k Fε,i (tuε,i ), for t ∈ R+ .

(10.240)

The condition k R (a) implies that the following majorisation relation similar to (10.240) takes place, for A ∈ BZ, i ∈ k X: k q0,i (t,

A) ≤ 1 − k F0,i (t), for t ∈ R+ .

(10.241)

¯ 1,i, i ∈ k X Note also that the choice of the distribution functions k Fε,i j (·), j ∈ k Y according to the relation (10.220) does not affect the condition k R, since the functions k qε,i (· k uε,i, A), A ∈ BZ, i ∈ k X do not depend on the above distribution functions. The following lemma takes place. Lemma 10.24 The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , k M, and R, assumed to be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)), entail the fulfilment of the condition R for the multi-alternating regenerative processes (k ξε (t), k ηε (t)) in the form of condition k R, where: (i) The limiting function k q0,i (·, A) appearing in the condition k R is defined for ˜ 1,i , or by the relation (10.247), if A ∈ BZ, i ∈ k X by the relation (10.244), if k  Y ˜ 1,i, w0,k,i ∈ (0, ∞). ˜ k ∈ Y1,i, w0,k,i = 0, or by the relation (10.262), if k ∈ Y (ii) The class of sets Γ appearing in the condition k R coincides with the class of sets Γ appearing in the condition R. (iii) The set of locally uniform convergence U[k q ·,i (· k u ·,i, A)] appearing in the ˜ 1,i , condition k R is defined for A ∈ Γ, i ∈ k X by the relation (10.245), if k  Y ˜ 1,i, w0,k,i = 0, or by the relation (10.281), if or by the relation (10.248), if k ∈ Y ˜ 1,i, w0,k,i ∈ (0, ∞). k∈Y (iv) The normalisation functions k u ·,i, i ∈ k X appearing in the condition k R are given by the relation (10.215). Proof The relations (10.210) and (10.215) imply that the following relation takes place, for t ≥ 0, A ∈ BZ and i ∈ k X: k qε,i (t k uε,i ,

A) = Pi {k ξε (t k uε,i ) ∈ A, k ζε,1 > t k uε,i } = Pi { ξ˜ε (t u˜ε,i ) ∈ A, ζ˜ε,1 > t u˜ε,i } ∫ t Pk { ξ˜ε (t − s)u˜ ε,i ∈ A, ζ˜ε,1 > (t − s)u˜ ε,i }Q˜ ε,ik (u˜ ε,i ds) + 0

= q˜ε,i (t u˜ε,i, A) + q˜ε,ik (t u˜ε,i, A),

(10.242)

238

10 Perturbed MARP

where ∫ q˜ε,ik (t u˜ε,i, A) =

t u˜ ε, i

0

∫ =

0

t

q˜ε,k ((t u˜ε,i − s)Q˜ ε,ik (ds)

q˜ε,k ((t − s)u˜ε,i, A)Q˜ ε,ik (u˜ε,i ds),

(10.243)

and the transition probabilities Q˜ ε,ik (s) are defined by the relation (10.68). ˜ 1,i , (2) For each i ∈ k X, three possible cases should be considered: (1) k  Y ˜ ˜ k ∈ Y1,i, w0,k,i = 0, (3) k ∈ Y1,i, w0,k,i ∈ (0, ∞). Here, w0,k,i is the parameter appearing in the condition k M1 . First, let us assume that: ˜ 1,i . (1) k  Y In this case, the condition k R holds, with the limiting functions k q0,i (t, A), t ≥ 0, A ∈ BZ given by the following relation: k q0,i (t,

A) = q˜0,i (t, A), t ≥ 0, A ∈ BZ

(10.244)

and the corresponding set of locally uniform convergence for the functions k qε,i (·, A) is given by the following relation, for A ∈ Γ: U[k q ·,i (· k u ·,i, A)] = U[q˜ ·,i (· u˜ ·,i, A)].

(10.245)

Here, Γ is the class of sets appearing in the condition R. Indeed, in this case, the probability p˜ε,ik = Q˜ ε,ik (∞) = 0, ε ∈ (0, 1] and, thus, for t ≥ 0, A ∈ BZ and ε ∈ (0, 1], k qε,i (t k uε,i ,

A) = q˜ε,i (t u˜ε,i, A).

(10.246)

The relation (10.246) implies that the condition k R for the functions k qε,i (t k uε,i, A) ˜ for the functions q˜ε,i (t u˜ε,i, A). is, just, a reduced form of the condition R Second, let us assume that: ˜ 1,i, w0,k,i = 0. (2) k ∈ Y In this case, the condition k R holds, with the limiting functions k q0,i (t, A), t ≥ 0, A ∈ BZ given by the following relation: k q0,i (t,

A) = q˜0,i (t, A), t ≥ 0, A ∈ BZ

(10.247)

and the set of locally uniform convergence for functions k qε,i (· k uε,i, A) is given by the following relation, for A ∈ Γ: U[k q ·,i (· k u ·,i, A)] = U[q˜ ·,i (· u˜ ·,i, A)] ∩ C[ F˜0,ik (·)] ∩ (0, ∞).

(10.248)

Here, as above, Γ is the class of sets appearing in condition R. Indeed, in this case, the function k qε,i (t k uε,i, A) is given by the relation (10.242). The following inequality takes place, for t ≥ 0, A ∈ BZ and ε ∈ (0, 1]:

10.3 MARP with reduced modulating SMP

∫ 0 ≤ q˜ε,ik (t u˜ε,i, A) =

t

q˜ε,k ((t − s)u˜ε,i, A)Q˜ ε,ik (u˜ε,i ds)

0

∫ ≤ ∫

t

q˜ε,k ((t − s)u˜ε,i, Z)Q˜ ε,ik (u˜ε,i ds)

0 t

=

Pk { ζ˜ε,1 > (t − s)u˜ ε,i }Q˜ ε,ik (u˜ ε,i ds)

0

∫ ≤

0

t

239

(1 − F˜ε,k ((t − s)u˜ε,i ))Q˜ ε,ik (u˜ε,i ds)

= q˜ε,ik (t u˜ε,i, Z).

(10.249)

˜ 1,i , then, According to the conditions I˜ 1 and J˜ 1 , if k ∈ Y Q˜ ε,ik (·u˜ε,i ) = F˜ε,ik (·u˜ε,i ) p˜ε,ik ⇒ F˜0,ik (·) p˜0,ik = Q˜ 0,ik (·) as ε → 0.

(10.250)

Also, the conditions I˜ 1 , J˜ 1 and the relation (10.85) imply that, F˜ε,k (·u˜ε,k ) ⇒ F˜0,k (·) as ε → 0.

(10.251)

Let us choose an arbitrary δ > 0. There is always sk,δ > 0, which is the continuity point for the distribution function F˜0,k (·), such that 1 − F˜0,k (sk,δ ) ≤ δ. u˜ If w0,k,i = 0, then u˜ ε,ε,ki → ∞ as ε → 0. Thus, for any 0 ≤ s < t < ∞ and δ > 0, u˜

there is εi,k,s,t,δ ∈ (0, 1], such that (t − s) u˜ ε,ε,ki ≥ sk,δ , for ε ∈ (0, εi,k,s,t,δ ]. Using the above remarks, we get the following relation, for 0 ≤ s < t < ∞: lim (1 − F˜ε,k ((t − s)u˜ε,i ))

ε→0

= lim (1 − F˜ε,k ((t − s) ε→0

u˜ε,i u˜ε,k )) u˜ε,k

≤ lim (1 − F˜ε,k (sk,δ u˜ε,k )) = 1 − F˜0,k (sk,δ ) ≤ δ. ε→0

(10.252)

Due to the arbitrary choice of δ > 0, it follows from the relation (10.252) that, for 0 ≤ s < t < ∞, (10.253) (1 − F˜ε,k ((t − s)u˜ε,i ) → 0 as ε → 0. Moreover, due to the monotonicity of the function (1 − F˜ε,k ((t − s)u˜ε,i )), s ∈ [0, t], it follows from the relation (10.253) that, for every t > 0, the above functions converge locally uniformly to the function 0(s) ≡ 0 at each point s ∈ [0, t), i.e., us (1 − F˜ε,k ((t − ·)u˜ε,i )) −→ 0(s) ≡ 0 as ε → 0.

(10.254)

Note that there is no guarantee that the locally uniform convergence of the functions (1 − F˜ε,k ((t − s), s ∈ [0, t] takes place for the point s = t. Recall that by C[ f (·)] we denote the set of continuity points of the function f (·).

240

10 Perturbed MARP

If t ∈ C[F˜0,ik (·)], then F˜0,ik ({t}) = Q˜ 0,ik ({t}) = 0. By the above remarks, the relations (10.250), (10.254), and Lemma B.2, the following relation takes place, for t ∈ C[F˜0,ik (·)] ∩ (0, ∞): ∫ t (1 − F˜ε,k ((t − s)u˜ε,i )Q˜ ε,ik (ds) q˜ε,ik (t u˜ε,i, Z) = 0 ∫ t 0(s)Q˜ 0,ik (ds) = 0 as ε → 0. (10.255) → 0

The relations (10.249) and (10.255) obviously imply that, for A ∈ BZ and t ∈ C[F˜0,ik (·)] ∩ (0, ∞), (10.256) q˜ε,ik (t u˜ε,i, A) → 0 as ε → 0. Let us choose an arbitrary 0 ≤ aε → 1 as ε → 0. It is easy to see that it is possible to repeat the above proof of relation (10.256)  = a u˜ using the normalisation function u˜ε,i ε ε,i instead of the normalisation function u˜ε,i , and thus get the following modified variant of the relation (10.256), for A ∈ BZ and t ∈ C[F˜0,ik (·)] ∩ (0, ∞): q˜ε,ik (taε u˜ε,i, A) → 0 as ε → 0.

(10.257)

Let us now choose an arbitrary 0 ≤ tε → t as ε → 0 and define aε = tε /t. The relation (10.257) takes, in this case, the following form, for A ∈ BZ and t ∈ C[F˜0,ik (·)] ∩ (0, ∞): q˜ε,ik (tε u˜ε,i, A) = q˜ε,ik (taε u˜ε,i, A) → 0 as ε → 0.

(10.258)

This relation implies that the following relation of locally uniform convergence takes place, for A ∈ BZ and t ∈ C[F˜0,ik (·)] ∩ (0, ∞): ut

q˜ε,ik (·u˜ε,i, A) −→ 0(·) ≡ 0 as ε → 0.

(10.259)

The relations (10.242), (10.259), and the asymptotic relation given in the condition ˜ obviously imply that, for s ∈ U[k q0,i (· k u ·,i, A)], A ∈ Γ, R k qε,i (· k uε,i,

A) = q˜ε,i (·u˜ε,i, A) + q˜ε,ik (·u˜ε,i, A) us

−→ q˜0,i (·, A) as ε → 0,

(10.260)

where the set U[k q ·,i (· k u ·,i, A)] is given by the relation (10.248). ¯ F˜0,ik (·)]) = 0, ˜ (c) m(U¯ [q˜ ·,i (· u˜ ·,i, A)]) = 0. Also, m(C[ According to the condition R ˜ since C[F0,ik (·)] is at most countable set. Therefore ¯ F˜0,ik (·)] ∪ {0}) = 0. m(U¯ [k q ·,i (· k u ·,i, A)]) = m(U¯ [q˜ ·,i (· u˜ ·,i, A)] ∪ C[

(10.261)

˜ (a) and k R (a) coincide, since the function k q0,i (t, A) = The conditions R ˜ (d) and k R (d) q˜0,i (t, A), t ∈ R+, A ∈ Γ. By the same reason, the conditions R

10.3 MARP with reduced modulating SMP

241

also coincide. Also, the relations (10.260) and (10.261) imply that the conditions k R (a) and (b) are satisfied. Third, let us assume that: ˜ 1,i, w0,k,i ∈ (0, ∞). (3) k ∈ Y In this case the condition k R is satisfied, with the limiting functions k q0,i (t, A), t ≥ 0, A ∈ BZ given by the following relation: ∫ t −1 q˜0,k ((t − s)w0,k,i , A)Q˜ 0,ik (ds) (10.262) k q0,i (t, A) = q˜0,i (t, A) + 0

and the corresponding sets of asymptotic uniform convergences U[k q0,i (· k u ·,i, A)], A ∈ Γ given below by relation (10.281). Here, as above, Γ is the class of sets appearing in the condition R. By the condition J˜ 1 , the distribution function F˜0,ik (·) does not have a singular component and, thus, can be represented in the form, F˜0,ik (·) = q˜ik,a F˜0,ik,a (·) + q˜ik,d F˜0,ik,d (·), where F˜0,ik,a (·) is an absolutely continuous distribution function, F˜0,ik,d (·) is a discrete distribution function, and q˜ik,a, q˜ik,d ≥ 0, q˜ik,a + q˜ik,d = 1. Let si,k,1, si,k,2, . . . be discontinuity points of the distribution function F˜0,ik,d (s) and Si,k = {si,k,0 = 0, si,k,1, , si,k,2, . . .}. The set Si,k is at most finite or countable set. Let, also, U[q˜ ·,k (· u˜ ·,k , A)], A ∈ Γ be the sets of locally uniform convergence ˜ (for the state i = k). According to the condition R ˜ the appearing in the condition R Lebesgue measure m(U¯ [q˜ ·,k (· u˜ ·,k , A)]) = 0, for any A ∈ Γ. For A ∈ Γ and u > 0, we introduce sets, U[q˜ ·,k (·u u˜ ·,k , A)] = {s ≥ 0 : su ∈ U[q˜ ·,k (· u˜ ·,k , A)]} ∪ (v, ∞),

(10.263)

¯ [q˜ ·,k (·u u˜ ·,k , A)] = {s ≥ 0 : su ∈ U¯ [q˜ ·,k (· u˜ ·,k , A)]}. U

(10.264)

and Obviously, m(U¯ [q˜ ·,k (·u u˜ ·,k , A)]) = 0.

(10.265)

u˜ε,i −1 → w0,k,i ∈ (0, ∞) as ε → 0. u˜ε,k

(10.266)

If w0,k,i ∈ (0∞), then

˜ and the relation (10.266) imply that, for A ∈ Γ and s ∈ The condition R

−1 u˜ , A)], U[q ·,k (· w0,k,i ·,k

q˜ε,k (·u˜ε,i, A) = q˜ε,k (· us

u˜ε,i u˜ε,k , A) u˜ε,k

−1 −→ q˜0,k (·w0,k,i , A) as ε → 0.

(10.267)

For A ∈ Γ and ε ∈ [0, 1], we continue the function q˜ε,k (t, A), t ≥ 0, as q˜ε,k (t, A) = 0, for t < 0.

242

10 Perturbed MARP

For A ∈ Γ and u, v > 0, we introduce sets, U[q˜ ·,k ((v − ·)u u˜ ·,k , A)]

= {s ∈ [0, v] : (v − s) ∈ U[q˜ ·,k (·u u˜ ·,k , A)]} ∪ (v, ∞) = {s ∈ [0, v] : (v − s)u ∈ U[q˜ ·,k (· u˜ ·,k , A)]} ∪ (v, ∞)

(10.268)

and ¯ [q˜ ·,k ((v − ·)u u˜ ·,k , A)] U = {s ∈ [0, v] : (v − s) ∈ U¯ [q˜ ·,k (·u u˜ ·,k , A)]} = {s ∈ [0, v] : (v − s)u ∈ U¯ [q˜ ·,k (· u˜ ·,k , A)].

(10.269)

Obviously, for A ∈ Γ and u, v > 0, m(U¯ [q˜ ·,k ((v − ·)u u˜ ·,k , A)]) = m(U¯ [q˜ ·,k (·u u˜ ·,k , A)] ∩ [0, v]) ≤ m(U¯ [q˜ ·,k (·u u˜ ·,k , A)]) = 0.

(10.270)

˜ and the relation Let 0 ≤ tε → t > 0 as ε → 0. This relation, the condition R, −1 (10.266) imply that, for s ∈ U[q ·,k ((t − ·)w0,k,i u˜ ·,k , A)], A ∈ Γ, us

−1 q˜ε,k ((tε − ·)u˜ε,i, A) −→ q˜0,k ((t − ·)w0,k,i , A) as ε → 0.

(10.271)

It was noted above that for any set S ∈ B+ such that m(S) = 0 and s ≥ 0, u > 0, the set T = {t > 0 : (t − s)u ∈ S} has Lebesgue measure m(T) = 0. For A ∈ Γ, we introduce sets, −1 Ti,k, A = ∩s˜i, k, n ∈S˜i, k {t > 0 : (t − si,k,n )w0,k,i ∈ U[q˜ ·,k (· u˜ ·,k , A)]} −1 + si,k,n ), = ∩si, k, n ∈Si, k (U[q˜ ·,k (· u˜ ·,k , A)]/w0,k,i

(10.272)

and −1 T¯i,k, A = ∪si, k, n ∈Si, k {t > 0 : (t − si,k,n )w0,k,i ∈ U¯ [q˜ ·,k (· u˜ ·,k , A)]} −1 = ∪si, k, n ∈Si, k (U¯ [q˜ ·,k (· u˜ ·,k , A)]/w0,k,i + si,k,n ),

(10.273)

where A/u + s denotes the set {a/u + s : a ∈ A}, for A ∈ B+, s ≥ 0, u > 0. By the above remarks, for A ∈ Γ, m(T¯i,k, A) = 0.

(10.274)

−1 ∈ U[q˜ (· u˜ , A)], for all s Also, if t ∈ Ti,k, A, then (t − si,k,n )w0,k,i ·,k ·,k i,k,n ∈ Si,k and, therefore, for any 0 ≤ tε → t as ε → 0. u s i, k, n

−1 q˜ε,k ((tε − ·)u˜ε,i, A) −→ q˜0,k ((t − ·)w0,k,i , A) as ε → 0.

(10.275)

10.3 MARP with reduced modulating SMP

243

By the definition of the sets T¯i,k, A, the probability Fˆ0,ik,d (T¯i,k, A) = 0. Also, ˆ F0,ik,a (T¯i,k, A) = 0, since m(T¯i,k, A) = 0. Therefore, F˜0,ik (T¯i,k, A) = 0. Let us now define a function q˜0,ik (t, A), for t ∈ R+, A ∈ BZ , ∫ ∞ −1 q˜0,ik (t, A) = q˜0,k ((t − s)w0,k,i , A)Q˜ 0,ik (ds), (10.276) 0

and, for A ∈ Γ, sets,

U[q˜ ·,ik (· u˜ ·,i, A)] = Ti,k, A .

(10.277)

By the above remarks, the relations (10.143), (10.157), and Lemma B.2, the following relation takes place, for A ∈ Γ and t ∈ U[q˜ ·,ik (· u˜ ·,i, A)], and any 0 ≤ tε → t as ε → 0: ∫ tε q˜ε,k ((tε − s)u˜ε,i, A)Q˜ ε,ik (u˜ε,i ds) q˜ε,ik (tε u˜ε,i, A) = 0 ∫ ∞ = q˜ε,k ((tε − s)u˜ε,i, A)Q˜ ε,ik (u˜ε,i ds) ∫0 ∞ −1 → q˜0,k ((t − s)w0,k,i , A)Q˜ 0,ik (ds) 0

= q˜0,ik (t, A) as ε → 0.

(10.278)

The relation (10.278) means that, for A ∈ Γ and t ∈ U[q˜ ·,ik (· u˜ ·,i, A)], ut

q˜ε,ik (·u˜ε,i, A) −→ q˜0,ik (·, A) as ε → 0.

(10.279)

Let us now define a function k q0,ik (t, A), for t ∈ R+, A ∈ BZ , k q0,i (t,

A) = q˜0,i (t, A) + q˜0,ik (t, A) ∫ t −1 q˜0,k ((t − u)w0,k,i , A)Q˜ 0,ik (du), = q˜0,i (t, A) +

(10.280)

0

and, for A ∈ Γ, sets, U[k q ·,i (· k u ·,i, A)] = U[q˜ ·,i (· u˜ ·,i, A)] ∩ U[q˜ ·,ik (· u˜ ·,i, A)].

(10.281)

˜ (c) and the relations (10.274), (10.277) imply that, for A ∈ Γ, The condition R m(U¯ [k q ·,i (· k u ·,i, A)]) = m(U¯ [q˜ ·,i (· u˜ ·,i, A)] ∪ U¯ [q˜ ·,ik (· u˜ ·,i, A)] = 0.

(10.282)

˜ (b) and the The relation of locally uniform convergence given in the condition R relation (10.279) imply that, for t ∈ U[k q ·,i (· k u ·,i, A)], A ∈ Γ, k qε,i (· k uε,i,

A) = q˜ε,i (·u˜ε,i, A) + q˜ε,ik (·u˜ε,i, A) ut

−→ q˜0,i (·, A) + q˜0,ik (·, A) = k q0,i (·, A) as ε → 0.

(10.283)

244

10 Perturbed MARP

Here, as above, Γ is the class of sets appearing in the condition R. The function k q0,i (t, A), t ∈ R+, A ∈ BZ , given by relation (10.280), belongs to the ˜ the functions q˜0,i (t, A), t ∈ R+, A ∈ class P[Z], since, according to the condition R, −1 BZ and q˜0,k ((t − u)w0,k,i, A), u ∈ R+, A ∈ BZ belong to the class P[Z]. Let us check that the condition k R (a) is satisfied, i.e., the function k q0,i (t, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − k F0,i (t), t ∈ R+ given by the relation (10.223). Lemma 10.10, the relations (10.215), (10.239), (10.283), and the inclusion relation Z ∈ Γ imply that the following relation holds for t ∈ C[k F0,i (·)] ∩ U[k q ·,i (· k u ·,i, Z)]: 1 − k Fε,i (t u˜ε,i ) = k qε,i (t u˜ε,i, Z) → 1 − k F0,i (t) = k q0,i (t, Z) as ε → 0.

(10.284)

The set C[k F0,i (·)] is at most countable and, therefore, m(U¯ [k q ·,i (· k u ·,i, Z)]) = 0. Thus, ¯ k F0,i (·)] ∪ U¯ [k q ·,i (· k u ·,i, Z)]) = 0. (10.285) m(C[ This relation implies that the set C[k F0,i (·)] ∩ U[k q ·,i (· k u ·,i, Z)] is dense in the interval [0, ∞). Function 1 − k F0,i (t) is continuous from the right on the interval [0, ∞). Function k q0,i (t, Z) is also continuous from the right on the interval [0, ∞). Indeed, according to Lemma 10.10, the functions q˜0,i (·, Z) = 1 − F˜0,i (·), i ∈ X are continuous from the right on the interval [0, ∞). Also, according to the relation (10.280), ∫ t −1 q˜0,k ((t − u)w0,k,i , Z)Q˜ 0,ik (du) k q0,i (t, Z) = q˜0,i (t, Z) + 0

= 1 − F˜0,i (t) + F˜0,ik (t) p˜0,ik ∫ t −1 − )F˜0,ik (du) p˜0,ik . F˜0,k ((t − u)w0,k,i

(10.286)

0

The distribution functions F˜0,i∫(t) and F˜0,ik (t) are continuous from the right on the t −1 ) F˜ interval [0, ∞). The convolution 0 F˜0,k ((t − u)w0,k,i 0,ik (du) is also a distribution function and, thus, it is continuous from the right on the interval [0, ∞). Since the functions 1 − k F0,i (t) and k q0,i (t, Z) coincide at the points of the subset of the interval [0, ∞), dense in this interval, and are right continuous, these function coincide, i.e., (10.287) k q0,i (t, Z) = 1 − k F0,i (t), for t ≥ 0. It remains to prove that the function k q0,i (·, A) given by the relation (10.280) is continuous almost everywhere with respect to the Lebesgue measure on B+ , for every A ∈ Γ.

10.3 MARP with reduced modulating SMP

245

As above, we use the notation C[q˜0,i (·, A)] for the set of continuity points for the ˜ is satisfied, function q˜0,i (·, A), for A ∈ Γ, i ∈ X. By Lemma 10.10 the condition R and, thus, for A ∈ Γ, i ∈ X, ¯ q˜0,i (·, A)]) = 0. m(C[

(10.288)

Therefore, it is enough to prove that the function q˜0,ik (·, A) given by the relation (10.276) is continuous almost everywhere with respect to Lebesgue measure on B+ , for every A ∈ Γ. For A ∈ Γ, we introduce sets, −1 Ri,k, A = ∩si, k, n ∈Si, k {t > 0 : (t − si,k,n )w0,k,i ∈ C[q˜0,k (·, A)]} −1 = ∩si, k, n ∈Si, k (C[q˜0,k (·, A)]/w0,k,i + si,k,n ),

(10.289)

and −1 ¯ q˜0,k (·, A)]} R¯i,k, A = ∪si, k, n ∈Si, k {t > 0 : (t − si,k,n )w0,k,i ∈ C[ −1 ¯ = ∪s + si,k.n ). ∈S (C[q˜0,k (·, A)]/w i, k, n

i, k

0,k,i

(10.290)

By the above remarks, for A ∈ Γ, m( R¯i,k, A) = 0.

(10.291)

−1 ∈ C[q˜ (·, A)], for all s Also, if t ∈ Ri,k, A, then (t − si,k,n )w0,k,i 0,k i,k,n ∈ Si,k and, therefore, for any 0 ≤ tε → t as ε → 0. −1 −1 q˜0,k ((tε − si,k,n ))w0,k,i , A) → q0,k ((t − si,k,n ))w0,k,i , A) as ε → 0.

(10.292)

By the definition of the sets R¯i,k, A, the probability F˜0,ik,d ( R¯i, A) = 0. Also, ˜ F0,ik,a ( R¯i,k, A) = 0, since m( R¯i,k, A) = 0. Therefore, F˜0,ik ( R¯i,k, A) = 0 and Q˜ 0,ik ( R¯i,k, A) = 0. By the above remarks and the Lebesgue theorem, the following relation takes place, for A ∈ Γ and t ∈ Ri,k, A, and any 0 ≤ tε → t as ε → 0: ∫ ∞ −1 q˜0,ik (tε u˜0,i, A) = q0,k ((tε − s)w0,k,i , A)Q˜ 0,ik (ds) 0 ∫ ∞ −1 → q0,k ((t − s)w0,k,i , A)Q˜ 0,ik (ds) 0

= q˜0,ik (t, A) as ε → 0. The proof of Lemma 10.24 is complete.

(10.293) 

246

10 Perturbed MARP

10.3.2 Modified Procedure of One-State Reduction of Phase Space for Modulating Semi-Markov Processes 10.3.2.1 Multi-Alternating Regenerative Processes with Reduced Phase Space for Modulating Semi-Markov Processes with Partially Removed Virtual Transitions. Let us assume that ε ∈ (0, 1] and the conditions G1 and H1 hold. Let us choose some state k ∈ X and introduce the reduced phase space k X = X \ {k}. Let us introduce, for ε ∈ (0, 1], the set of states of “entry” into the state k ∈ X for the semi-Markov process ηε (t), ˆ ε,k = {i ∈ k X : pε,ik > 0}. Y

(10.294)

The following simple lemma takes place. Lemma 10.25 Let the conditions G1 and H1 be satisfied for the semi-Markov proˆ ε,k = Y ˆ 1,k , ε ∈ (0, 1], for k ∈ X and, cesses ηε (t). Then, sets Y ˆ 1,k = {i ∈ k X : k ∈ Y1,i }. Y

(10.295)

Let K ⊆ X be a non-empty subset of the phase space X such that, ˆ 1,k ∪ {k} ⊆ K. Y

(10.296)

Let us define the following stopping times for the Markov chain η˜ε,K,n , for r = 0, 1, . . .: (10.297) k αε,K [r] = min(n > r : η˜ε,K,n ∈ k X). By the definition, k αε,K [r] is the first after r time of hitting into the reduced phase space k X by the Markov chain η˜ε,K,n . Since the Markov chain η˜ε,K,n does not make virtual transitions of the form i → i, for i ∈ K, the following relation takes place, for r = 0, 1, . . .:  r + 1 if η˜ε,K,r+1 ∈ k X, (10.298) k αε,K [r] = r + 2 if η˜ε,K,r+1 = k. Let us also define the following sequential stopping times for Markov chains η˜ε,K,n : k βε,K,n

= k αε [k βε,K,n−1 ], n = 1, 2, . . . , where k βε,K,0 = I(η˜ε,K,0 = k). (10.299)

Let us now construct a new Markov renewal process (k ηε,K,n, k κε,K,n ), n = 0, 1, . . ., with the phase space k X × [0, ∞), (k ηε,K,n, k κε,K,n )

10.3 MARP with reduced modulating SMP

=

for n = 0, (η˜ε,K, k βε,K,0 , 0) k βε,K, n (η˜ε,K, k βε,K, n , l= κ ˜ ) ε,K,l for n = 1, 2, . . . k β ε,K, n−1 +1

247

(10.300)

and define the corresponding reduced modulating semi-Markov process, k ηε,K (t)

= k ηε,K, k νε,K (t), t ≥ 0,

(10.301)

where, k ζε,K,n

= k κε,K,1 + · · · + k κε,K,n, n = 1, 2, . . . , k ζε,K,0 = 0,

(10.302)

are the corresponding moments of jumps, and, k νε,K (t)

= max(n ≥ 1 : k ζε,K,n ≤ t),

(10.303)

is the number of jumps in an interval [0, t] for the above semi-Markov process. By the definition of the semi-Markov process k ηε,K (t), the following relation holds, for i ∈ X: Pi {k ηε,K (t) ∈ k X, t ≥ 0} = 1. (10.304) The relation (10.304) makes it possible to consider k ηε,K (t) as a semi-Markov process with the reduced phase space k X and refer to k ηε,K (t) as a reduced modulating semi-Markov process with partially removed virtual transitions. Let us consider a two component stochastic process, (k ξε,K (t), k ηε,K (t)) = (ξε (ζ˜ε,K, k βε,K,0 + t), k ηε,K (t)), t ≥ 0.

(10.305)

This is a multi-alternating regenerative process. It has a regenerative component the same as the original multi-alternating regenerative process (ξε (t), ηε (t)), but has a new modulating semi-Markov process k ηε,K (t). The new multi-alternating regenerative process with reduced modulating semiMarkov process (k ξε,K (t), k ηε,K (t)) takes the following form: (k ξε,K (t), k ηε,K (t)) = (ξε (ζ˜ε,K, k βε,K,0 + t), k ηε,K (t)), t ≥ 0.

(10.306)

Due to the relation (10.296), the transition probabilities for the Markov renewal process (k ηε,K,n, k κε,K,n ) take the following form, for t ≥ 0, i, j ∈ k X:  Q ε,i j (t) for i ∈ k X \ K, j ∈ k X, (10.307) k Q ε,K,i j (t) = Q˜ ε,i j (t) + Q˜ ε,ik (t) ∗ Q˜ ε,k j (t) for i ∈ k X ∩ K, j ∈ k X. The transition probabilities for the embedded Markov chain k ηε,K,n are given by the following relation, for i, j ∈ k X:  for i ∈ k X \ K, j ∈ k X, pε,i j (10.308) p = k ε,K,i j p˜ε,i j + p˜ε,ik p˜ε,k j for i ∈ k X ∩ K, j ∈ k X.

248

10 Perturbed MARP

Let us also introduce the distribution functions of sojourn times for the semiMarkov process k ηε,K (t),

(10.309) k Q ε,K,i j (t), t ≥ 0, i ∈ k X. k Fε,K,i (t) = j∈ kX

10.3.2.2 Perturbation Conditions for Multi-Alternating Regenerative Processes with Reduced Phase Space for Modulating Semi-Markov Processes with Partially Removed Virtual Transitions. Analogues of the conditions k G1 and k H1 take for the semi-Markov processes k ηε,K (t) the following forms:  k GK :

(a) k pε,K,i j > 0, ε ∈ (0, 1] or k pε,K,i j = 0, ε ∈ (0, 1], for every i, j ∈ k X, (b) F k ε,K,i (0) = 0, for i ∈ k X, ε ∈ (0, 1]

and  k HK : For any i,

j ∈ k X, there exist k nK,i j ≥ 1 and a chain of states i = k rK,0 , k rK,1, . . ., r = j, all from space X, such that k K, k nK, i j k 1≤l ≤ k nK, i j k p1,K, k rK, l−1 k rK, l > 0.

The following lemma in a direct corollary of Lemma 9.141 and an analogue of Lemma 10.20. Lemma 10.26 The conditions G1 and H1 , assumed to be satisfied for the semiMarkov processes ηε (t), entail the fulfilment of conditions G1 and H1 for the semiMarkov processes k ηε,K (t) in the form of conditions k GK and k HK . The following sets were introduced in the relations (8.81)1 and (8.82)1 :  Y1,i for i ∈ k X \ K, (10.310) k Y1,K,i = { j ∈ k X : k p1,K,i j > 0} = k Y1,i for i ∈ k X ∩ K. Analogues of the conditions I1 and IH,1 take for the semi-Markov processes the following forms:

k ηε,K (t)

 k IK : k pε,K,i j

→ k p0,K,i j as ε → 0, for i, j ∈ k X

and  k IH,K :

The functions k p ·,K,i j , j ∈ k Y1,K,i, i ∈ k X belong to the complete family of asymptotically comparable functions H appearing in the condition IH,1 .

Obviously, k p0,K,i j  is a stochastic matrix.  . Note also that the condition k IK is implied by the condition k IH,K The following lemma is a direct corollary of Lemma 9.171 and an analogue of Lemma 10.21. Lemma 10.27 The conditions G1 , H1 , I1 , and IH,1 , assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of the conditions I1 and IH,1 for  . the semi-Markov processes k ηε,K (t) in the form of conditions k IK and k IH,K

10.3 MARP with reduced modulating SMP

249

Analogues of the normalisation functions k uε,i, i ∈ X used in the condition k J take the following forms, for i ∈ k X:  uε,i for i ∈ k X \ K, (10.311) k uε,K,i = u˜ ε,K,i = u˜ε,i = p¯−1 u ε,ii ε,i for i ∈ k X ∩ K. Note that the conditions G1 and H1 imply that p¯ε,ii ∈ (0, 1], ε ∈ 0, 1], for i ∈ X and, thus, due to the above conditions and the condition J1 (c), for i ∈ k X, k uε,K,i

∈ (0, ∞), ε ∈ (0, 1].

(10.312)

Analogues of the conditions k LK and k LH,1 take for the semi-Markov processes k ηε,K (t) the following forms:  k LK : k uε,K,i

→ k u0,K,i ∈ (0, ∞] as ε → 0, for i ∈ k X

and  k LH,K : The functions k u ·,K,i, i

∈ k X belong to the complete family of asymptotically comparable functions H appearing in the condition LH,1 .

The following lemma is the direct corollary of Lemma 10.6 and an analogue of Lemma 10.22. Lemma 10.28 The conditions G1 , H1 and I1 , IH,1 , L1 , LH,1 , assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of the conditions L1 and LH,1 for the semi-Markov processes k ηε,K (t) in the form of conditions k LK and  k LH,K . Note, also, that the conditions G1 , H1 , I1 , L1 , and the relation (10.78) imply that limits k u0,K,i, i ∈ k X appearing in the condition k L1 take the following forms: k u0,K,i

=

⎧ if i ∈ k X \ K, ⎪ ⎨ u0,i ⎪ p¯−1 u ∈ (0, ∞) if p¯0,ii > 0, u0,i ∈ (0, ∞), i ∈ k X ∩ K, 0,ii 0,i ⎪ ⎪∞ if p¯0,ii = 0 or p¯0,ii > 0, u0,i = ∞, i ∈ k X ∩ K. ⎩

(10.313)

Analogues of the distribution functions k Fε,i j (u) and their Laplace transform given, respectively, by the relations (9.83)1 and (9.86)1 , take the following forms, for ε ∈ (0, 1] and j ∈ k Y1,K,i, i ∈ k X: k φε,i j (s),

⎧ Fε,i j (t) ⎪ ⎪ ⎪ ⎪ for t ≥ 0, j ∈ k Y1,K,i, i ∈ k X \ K, ⎪ ⎪ ⎨ ⎪ p˜ F (t) = F˜ε,i j (t) k pε,ε,iijj k Fε,K,i j (t) = k ε,i j ⎪ p˜ p˜ ⎪ ⎪ +F˜ε,ik (t) ∗ F˜ε,k j (t) ε,kipkε, iε,j k j , ⎪ ⎪ ⎪ ⎪ for t ≥ 0, j ∈ Y k 1,K,i, i ∈ k X ∩ K, ⎩ and

(10.314)

250

10 Perturbed MARP

⎧ φε,i j (s) ⎪ ⎪ ⎪ ⎪ for s ≥ 0, j ∈ k Y1,K,i, i ∈ k X \ K, ⎪ ⎪ ⎨ ⎪ p˜ ε, i j φ k ε,i j (s) = φ˜ε,i j (s) k pε, i j k φε,K,i j (s) = ⎪ p˜ p˜ ⎪ ⎪ +φ˜ε,ik (s)φ˜ε,k j (t) ε,kipkε, iε,j k j , ⎪ ⎪ ⎪ ⎪ for s ≥ 0, j ∈ Y k 1,K,i, i ∈ k X ∩ K. ⎩

(10.315)

Also, we can define the corresponding distribution function k Fε,K,i j (u) and its ¯ 1,K,i, i ∈ k X, Laplace transform k φε,K,i j (s), for ε ∈ (0, 1] and j ∈ k Y k Fε,K,i j (t)

= k Fε,K,i (t), t ≥ 0

∫

and k φε,K,i j (s) = k φε,K,i (s) =

∞

0

e−su k Fε,K,i (du), s ≥ 0.

(10.316)

(10.317)

¯ 1,K,i, i ∈ k X does not afNote that the above choice of k Fε,K,i j (t) for j ∈ k Y fect the transition probabilities k Q ε,K,i j (t) of the semi-Markov process k ηε,K (t) and analogues of the conditions k G1 , k H1 , k I1 , k IH,1 , k L1 , k LH,1 , J1 , and k K1 for the semi-Markov processes k ηε,K (t) (see, Sect. 10.1.1.1 and Remark 10.1). Analogues of the conditions k J1 and k J◦1 take for the semi-Markov processes k ηε,K (t) the following forms:  k JK : (a) k Fε,K,i j (· k uε,K,i )

⇒

k F0,K,i j (·)

as ε → 0, for j ∈ k Y1,K,i, i ∈ k X, (b)

k F0,K,i j (·) is a non-arithmetic distribution function without singular component,

for j ∈ k Y1,K,i, i ∈ k X, (c) k uε,K,i ∈ (0, ∞), ε ∈ (0, 1], for i ∈ k X

and ◦ k JK : (a) k φε,K,i j (s/ k uε,K,i )

→ k φ0,K,i j (s) as ε → 0, for s ≥ 0 and j ∈ k Y1,K,i, i ∈ (b) k φ0,K,i j (·) is the Laplace transform of non-arithmetic distribution function without singular component k F0,K,i j (·), for j ∈ k Y1,K,i, i ∈ k X, (c) k uε,K,i ∈ (0, ∞), ε ∈ (0, 1], for i ∈ k X. k X,

The conditions k GK , k HK , and k IK imply that, for i ∈ k X, k Y0,K,i

⊆ k Y1,K,i .

(10.318)

Below, we shall show that the normalisation functions k uε,K,i, i ∈ k X given by the relation (10.311) can be used in the conditions k JK and k JK◦ . The conditions k GK , k HK , k IK , k JK , and the relation (10.318) imply that, for i ∈ k X,

10.3 MARP with reduced modulating SMP k Fε,K,i (·k uε,K,i )



=

251 k Q ε,K,i j (· k uε,K,i )

j ∈ k Y1,K, i



=

k Fε,K,i j (· k uε,K,i ) k pε,K,i j

j ∈ k Y1,K, i

⇒



k F0,K,i j (·) k p0,K,i j

j ∈ k Y1,K, i



=

k F0,K,i j (·) k p0,K,i j

j ∈ k Y0,K, i

= k F0,K,i (·) as ε → 0.

(10.319)

The conditions k GK , k HK , k IK , k JK and relation (10.318) imply that, for i, j ∈ k X, k Q ε,K,i j (·)

= k Fε,K,i j (·) k pε,K,i j ⇒ k F0,K,i j (·) k p0,K,i j = k Q0,K,i j (·) as ε → 0.

(10.320)

⎧ for j ∈ Y0,i, i ∈ k X \ K, ⎪ ⎨ F0,i j (·)p0,i j ⎪ k Q 0,K,i j (·) = k F0,i j (·) k p0,i j for j ∈ k Y0,i, i ∈ k X ∩ K, ⎪ ⎪0 ¯ 0,i, i ∈ k X ∩ K. for j ∈ k Y ⎩

(10.321)

Note that

Lemma 9.201 implies that, under the condition k IH,1 , the following asymptotic ˆ formulated in Sect. 9.3.2.21 ) holds, for i, j ∈ X: relation (given in the condition L wε,K, j,i =

u˜ε,K, j → w0,K, j,i ∈ [0, ∞] as ε → 0. u˜ε,K,i

(10.322)

Since, u˜ε,K,i ≡ u˜ε,i , for i ∈ K, the limits w0,K,i, j = w0,i, j , for i, j ∈ K. Let us introduce set, W0,K = { j ∈ K : w0,K, j,i ∈ [0, ∞), i ∈ K}.

(10.323)

Obviously, for any non-empty set K ⊆ X, W0 ⊆ W0,K .

(10.324)

Let us introduce condition:  k MK :

k ∈ W0,K , i.e., limε→0

u˜ ε,K, k u˜ ε,K, i

= w0,K,k,i ∈ [0, ∞), for i ∈ K.

Obviously the condition k M1 is sufficient for holding of the condition k MK . The following lemma is an analogue of Lemmas 9.211 and 10.23. Lemma 10.29 The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J◦1 (J1 ), K1 , and k MK , assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of the condition J◦1 (J1 ) for the semi-Markov processes k ηε,K (t) in the form of condition ◦  k JK (k JK ), where:

252

10 Perturbed MARP

(i) The limiting Laplace transforms k φ0,i j (·), j ∈ k Y1,K,i, i ∈ k X appearing in the condition k J◦1 are given by the following relation: ⎧ φ0,i j (s) ⎪ ⎪ ⎪ ⎪ ⎪ for s ≥ 0, j ∈ k Y1,K,i, i ∈ k X \ K, ⎨ ⎪ k φ0,K,i j (s) = k φ0,i j (s) = φ˜0,i j (s) k q0,i j ⎪ ⎪ + φ˜0,ik (s)φ˜0,k j (w0,k,i s)(1 − k q0,i j ) ⎪ ⎪ ⎪ ⎪ for s ≥ 0, j ∈ k Y1,K,i, i ∈ k X ∩ K. ⎩ (ii) The normalisation functions k u ·,K,i, i ∈ (k JK ) are given by the relation (10.311).

kX

(10.325)

appearing in the condition k JK◦

Analogues of the expectations f˜ε,i j given by the relation (8.65)1 take the following forms: for ε ∈ (0, 1] and j ∈ k Y1,K,i, i ∈ k X, ∫ ∞ u k Fε,K,i (du) k fε,K,i j = 0

⎧ fε,i j ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ for j ∈ k Y1,K,i, i ∈ k X \ K, ⎪ = k fε,i j = fε,i j pε,i j + ( fε,ik ⎪ ⎪ + fε,k j )pε,ik pε,k j ⎪ ⎪ ⎪ ⎪ for j ∈ k Y1,K,i, i ∈ k X ∩ K. ⎩

(10.326)

An analogue of the condition k K1 takes for the semi-Markov processes k ηε,K (t) the following form:  k KK : (a) k fε,K,i ∫ j∞< ∞, j ∈ k Y1,K,i, i ∈ X, for every ε k f0,i j = 0 u k F0,K,i j (du) < ∞ as ε → 0, for j

∈ (0, 1], (b) k fε,K,i j / k uε,K,i → ∈ k Y1,K,i, i ∈ k X.

The following lemma is a corollary of Lemma 9.221 and an analogue of Lemma 10.24. Lemma 10.30 The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , and k MK , assumed to be satisfied for the semi-Markov processes ηε (t), entail the fulfilment of condition K1 for the semi-Markov processes k ηε,K (t) in the form of condition k KK , where: (i) The limiting expectations k f0,i j , j ∈ k Y1,K,i, i ∈ k X appearing in the condition  k KK are given by the following relation:

k f0,K,i j

⎧ f0,i j ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ for j ∈ k Y1,K,i, i ∈ k X \ K, ⎪ = k f0,i j = f˜0,i j k q0,i j ⎪ ⎪ + ( f˜0,ik + f˜0,k j w0,k,i )(1 − k q0,i j ) ⎪ ⎪ ⎪ ⎪ for j ∈ k Y1,K,i, i ∈ k X ∩ K. ⎩

(10.327)

(ii) The normalisation functions k u ·,K,i, i ∈ k X appearing in the condition k KK are given by the relation (10.311).

10.3 MARP with reduced modulating SMP

253

The functions k qε,K,i (t, A), which are analogues of functions qε,i (t, A) are defined, for ε ∈ (0, 1] by the following relation, for i ∈ k X: k qε,K,i (t,

A) = Pi {k ξε,K (t) ∈ A, k ζε,K,1 > t}  q (t, A) for t ≥ 0, i ∈ k X \ K, A ∈ BZ, = ε,i k qε,i (t, A) for t ≥ 0, i ∈ k X ∩ K, A ∈ BZ .

(10.328)

The relation (10.328) implies that the functions k qε,K,i (t k uε,K,i, A), t ∈ R+, A ∈ BZ belong to the class P[BZ ], for i ∈ k X, ε ∈ (0, 1]. Moreover, the function k qε,K,i (t, k uε,K,i, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − k Fε,Ki (t k uε,K,i ), for i ∈ k X and ε ∈ (0, 1], i.e., k qε,K,i (t k uε,K,i, Z)

= 1 − k Fε,K,i (t k uε,K,i ), for t ∈ R+ .

(10.329)

An analogue of the condition R takes for the multi-alternating regenerative processes (k ξε,K (t), k ηε,K (t)) the following form:  k RK :

There exist functions k q0,i (t, K, A), t ≥ 0, A ∈ BZ , i ∈ k X, which belong to the class P[BZ ], a class of set Γ ⊆ BZ , and Borel sets U[k q0,K,i (· k u ·,K,i, A)], A ∈ Γ, i ∈ k X such that: (a) the function k q0,K,i (t, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − k F0,K,i (t), t ∈ R+ , for i ∈ k X; us (b) the functions k qε,K,i (· k uε,K,i, A) −→ k q0,K,i (·, A) as ε → 0, for points s ∈ U[k q ·,K,i (· k u ·,K,i, A)], A ∈ Γ, i ∈ k X; (c) m(U¯ [k q ·,K,i (· k u ·,K,i, A)]) = 0, for A ∈ Γ, i ∈ k X; (d) the function k q0,K,i (·, A) is continuous almost everywhere with respect to the Lebesgue measure m(·) on B+ , for A ∈ Γ, i ∈ k X.

¯ 1,K,i, i ∈ Note also that the choice of the distribution functions k Fε,K,i j (·), j ∈ k Y  k X according to the relation (10.316) does not affect the condition k RK since the functions k qε,K,i (· k uε,K,i, A), i ∈ k X, A ∈ BZ do not depend on the above distribution functions. The following lemma is an analogue of Lemma 10.25. Lemma 10.31 The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , k MK , and R, assumed to be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)), entail the fulfilment of the condition R for the multi-alternating regenerative processes (k ξε,K (t), k ηε,K (t)) in the form of the condition k RK , where: (i) The limiting function k q0,K,i (·, A) appearing in condition k RK coincides with the function q0,i (·, A) appearing in the condition R, for A ∈ BZ, i ∈ i ∈ k X \ K, ˜ 1,i , or or is defined, for A ∈ BZ, i ∈ k X ∩ K, by the relation (10.244), if k  Y ˜ by the relation (10.247), if k ∈ Y1,i, w0,k,i = 0, or by the relation (10.262), if ˜ 1,i, w0,k,i ∈ (0, ∞). k∈Y

254

10 Perturbed MARP

(ii) The class of sets Γ appearing in condition k RK coincides with the class of sets Γ appearing in condition R. (iii) The set of locally uniform convergence U[k q ·,K,i (· k u ·,K,i, A)] appearing in condition k RK coincides with the set U[q ·,i (· u ·,i, A)] appearing in the condition R, for A ∈ BZ, i ∈ i ∈ k X \ K, or is defined for A ∈ BZ, i ∈ k X ∩ K by the relation ˜ 1,i, w0,k,i = 0, or by the ˜ 1,i , or by the relation (10.248), if k ∈ Y (10.245), if k  Y ˜ relation (10.281), if k ∈ Y1,i, w0,k,i ∈ (0, ∞). (iv) The normalisation functions k u ·,K,i, i ∈ k X appearing in condition k RK , are given by relation (10.311).

Chapter 11

Time–Space Aggregation of Regeneration Times for Perturbed Multi-Alternating Regenerative Processes

In this chapter, asymptotic recurrent algorithms of time–space aggregation of regeneration times for perturbed multi-alternating regenerative processes, based on total and partial removing of virtual transitions and reduction of phase space for modulating semi-Markov processes, are described. This chapter includes four sections. In Sect. 11.1, we present and compare multi-alternating regenerative processes of two types, with totally or partially removed virtual transitions and reduced phase space for modulating semi-Markov processes. In Sect. 11.2, we describe the asymptotic recurrent algorithm of time–space aggregation of regeneration times for perturbed multi-alternating regenerative processes based on total removing virtual transitions and reduction of phase space for the corresponding modulating semi-Markov processes. The outcomes of this algorithm are summarised in Theorem 11.1. In Sect. 11.3, we describe the asymptotic recurrent algorithm of time–space aggregation of regeneration times for perturbed multi-alternating regenerative processes based on partial removing virtual transitions and reduction of phase space for the corresponding modulating semi-Markov processes. The outcomes of this algorithm are summarised in Theorem 11.2. In Sect. 11.4, we compare the above two asymptotic recurrent algorithms of time– space aggregation of regeneration times for perturbed multi-alternating regenerative processes. In Lemma 11.1 we present results related to comparison of time compression factors and absorbing rate conditions used in the above algorithms. In Lemmas 11.2 and 11.3, we compare asymptotic communicative structure of modulating semi-Markov processes for embedded alternating regenerative processes constructed in the above algorithms.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0_11

255

256

11 Time-space aggregation of regeneration times

11.1 Multi-Alternating Regenerative Processes with Removed Virtual Transitions and Reduced Phase Space for Modulating Semi-Markov Processes In this section, we define multi-alternating regenerative processes of two types, with totally or partially removed virtual transitions and the phase space of modulating semi-Markov process reduced by one state.

11.1.1 Multi-Alternating Regenerative Processes with Totally or Partially Removed Virtual Transitions for Modulating Semi-Markov Processes Section 10.2.1 describes the procedure of total removal of virtual transitions of the form i → i (for all i ∈ X) from trajectories of modulating semi-Markov processes. This procedure transforms the original multi-alternating regenerative process (ξε (t), ηε (t)) into the multi-alternating regenerative process (ξ˜ε (t), η˜ε (t)) with totally removed virtual transitions. Section 10.2.2 describes the procedure of partial removal of virtual transitions of the form i → i (for i ∈ K, where K ⊆ X) from trajectories of the modulating semi-Markov processes. This procedure transforms the original multi-alternating regenerative process (ξε (t), ηε (t)) into the multi-alternating regenerative process (ξ˜ε,K (t), η˜ε,K (t)) with partially removed virtual transitions. Suppose now that the condition k M holds for a state k ∈ X, i.e., that the following relation holds: u˜ε,k ∈ [0, ∞), for i ∈ X, (11.1) w0,k,i = lim ε→0 u˜ ε,i where

u˜ε,i = p¯−1 ε,ii uε,i, for i ∈ X.

(11.2)

We introduce the sets, for k ∈ X, ˆ 1,k ∪ {k} = {i ∈ k X : p1,ik > 0} ∪ {k}. K[k] = Y

(11.3)

ˆ 1,k was defined by the relations (10.294) and (10.295). Recall that the set Y We use the abbreviated notation (k ξ˜ε (t), k η˜ε (t)) for the multi-alternating regenerative process (ξ˜ε,K[k] (t), η˜ε,K[k] (t)), which is the result of removing virtual transitions of the form i → i (for i ∈ K[k]) from the trajectories of the modulating semi-Markov process ηε (t) for the multi-alternating regenerative process (ξε (t), ηε (t)). In this context, it is also natural to use abbreviated notation for the conditions     ˜ ˜ 1 , H ˜ ˜ K[k] ˜ H,K[k] ˜ 1 , I˜  ˜ 1 , L G = kG = kH = k I˜ 1 , I˜ H,K[k] = k I˜ H,1 , L = kL = K[k] K[k] K[k]        ˜ ˜ ˜ ˜ ˜ ˜ ˜ = k J1 , K = k K1 , and R = kR . k LH,1 , J K[k]

K[k]

K[k]

11.1 MARP with reduced phase spaces of modulating SMP

257

Also, the following abbreviated notation is used for the normalisation functions (applied for the multi-alternating regenerative process (k ξ˜ε (t), k η˜ε (t))),  ¯ uε,i for i ∈ K[k],  (11.4) k u˜ ε,i = u˜ ε,K[k],i = u˜ε,i = p¯−1 u for i ∈ K[k]. ε,ii ε,i The process (k ξ˜ε (t), k η˜ε (t)) is an alternative for the process (ξ˜ε (t), η˜ε (t)). The procedure of total removal of virtual transitions of the form i → i (for all i ∈ X) from trajectories of the modulating semi-Markov process ηε (t) transforms the process (ξε (t), ηε (t)) into the process (ξ˜ε (t), η˜ε (t)). The procedure of partial removal of virtual transitions of the form i → i (for i ∈ K[k]) from trajectories of the modulating semi-Markov process ηε (t) transforms the process (ξε (t), ηε (t)) into the process (k ξ˜ε (t), k η˜ε (t)). It is useful to note that the process (k ξ˜ε (t), k η˜ε (t)) coincides with the process ˜ (ξε (t), η˜ε (t)), if the set K[k] = X.

11.1.2 Multi-Alternating Regenerative Processes with Reduced Phase Space of Modulating Semi-Markov Processes Section 10.3.1 describes the procedure of exclusion of a state k ∈ X from the phase space X for modulating semi-Markov processes with totally removed virtual transitions. This procedure transforms the multi-alternating regenerative process (ξ˜ε (t), η˜ε (t)) into the multi-alternating regenerative process (k ξε (t), k ηε (t)) with the reduced phase space k X. Section 10.3.2 describes the procedure of exclusion of a state k ∈ X from the phase space X for modulating semi-Markov processes with partially removed virtual transitions. This procedure transforms the multi-alternating regenerative process  (t), η˜  (t)) (for the case, where K[k] ⊆ K ⊆ X) into the multi-alternating (ξ˜ε,K ε,K regenerative process (k ξε,K (t), k ηε,K (t)) with the reduced phase space k X. In what follows, we consider the case when K = K[k] ⊆ X.

(11.5)

We use the abbreviated notation (k ξε (t), k ηε (t)) for the multi-alternating regenerative process (k ξε,K[k] (t), k ηε,K[k] (t)), which is the result of exclusion of state k from the phase space X of the modulating semi-Markov process k η˜ε (t) for the multi-alternating regenerative process (k ξ˜ε (t), k η˜ε (t)) = (ξ˜ε,K[k] (t), η˜ε,K[k] (t)).  holds for some state k ∈ X, i.e., Suppose now that the condition k MK[k]  = lim w0,k,i

 k u˜ ε,k

ε→0 k u˜  ε,i

∈ [0, ∞), for i ∈ K[k].

(11.6)

258

11 Time-space aggregation of regeneration times

In this context, it is also natural to use abbreviated notation for the conditions       = k G1 , k HK[k] = k H1 , k IK[k] = k I1 , k IH,K[k] = k IH,1 , k LK[k] = k L1 , GK[k]           k LH,K[k] = k LH,1 , k JK[k] = k J1 , k KK[k] = k K1 , k RK[k] = k R , and k MK[k] = k M . Also, the following reduced notations for normalisation functions (applied for the multi-alternating regenerative process (k ξε (t), k ηε (t))) are used,  uε,i for i ∈ k X \ K[k],  (11.7) u = u = k ε,i k ε,K[k],i u˜ε,i = p¯−1 u for i ∈ k X ∩ K[k]. ε,i ε,ii The process (k ξε (t), k ηε (t)) is an alternative to process (k ξε (t), k ηε (t)). The procedure of exclusion of state k from the phase space X of process η˜ε (t)) is used for transformation of process (ξ˜ε (t), η˜ε (t)) into the process (k ξε (t), k ηε (t)). The procedure of exclusion of state k from the phase space X of process k η˜ε (t) is used for transformation of process (k ξ˜ε (t), k η˜ε (t)) into process (k ξε (t), k ηε (t)). It is useful to note that the process (k ξε (t), k ηε (t)) coincides with the process (k ξε (t), k ηε (t)), if the set K[k] = X.

11.2 Time–Space Aggregation for Regeneration Times Based on Total Removing of Virtual Transitions This section describes an asymptotic recurrent algorithm for space-time aggregation for regeneration times for multivariate regenerative processes, based on total removal of virtual transitions from the trajectories of modulating semi-Markov processes and reduction of its phase space.

11.2.1 Algorithm of Time–Space Aggregation for Regeneration Times Based on Total Removing of Virtual Transitions 11.2.1.1 Notation. This algorithm is based on the recurrent application of the procedure of total removal of virtual transitions and the procedure of exclusion of one state from the phase space of modulating semi-Markov processes. These procedures are described in Sects. 10.2.1, 10.3.1, and 11.1. The key element for this algorithm is the procedure of successive exclusion of a sequence of states k¯ n = k1, k2, . . . , k n , for some 1 ≤ n ≤ m − 1, from the phase space X of the modulating semi-Markov processes ηε (t) (for the multi-alternating regenerative process (ξε (t), ηε (t))). This procedure results in the reduction of the phase space X to the space k¯ n X = X \ {k1, k2, . . . , k n }. We use the abbreviated notation (k¯ n ξε (t), k¯ n ηε (t)) for the multi-alternating regenerative process obtained by successive exclusion of the sequence of states k¯ n = k1, . . . , k n from the phase space X of the modulating semi-Markov process ηε (t).

11.2 Aggregation of regeneration times based on total removing of virtual transitions

259

Process (k¯ n ξε (t), k¯ n ηε (t)) is the result of the n-th step of the algorithm that includes two sub-steps. At the first sub-step, virtual transitions of the form i → i are totally (for all i ∈ k¯ n−1 X) removed from trajectories of the modulating semi-Markov process k¯ n−1 ηε (t) (for the multi-alternating regenerative process (k¯ n−1 ξε (t), k¯ n−1 ηε (t))). We use the abbreviated notation (k¯ n−1 ξ˜ε (t), k¯ n−1 η˜ε (t)) for the obtained multi-alternating regenerative process. At the second sub-step, the state k n is excluded from the phase space of the modulating semi-Markov process k¯ n−1 η˜ε (t), resulting in the multi-alternating regenerative process (k¯ n ξε (t), k¯ n ηε (t)). Note that the multi-alternating regenerative process (k¯ n−1 ξ˜ε (t), k¯ n−1 η˜ε (t)) has the phase space Z × k¯ n−1 X, while the multi-alternating regenerative process (k¯ n ξε (t), k¯ n ηε (t)) has the phase space Z × k¯ n X. In particular, we change the notation from (k1 ξε (t), k1 ηε (t)) and (k1 ξ˜ε (t), k1 η˜ε (t)), respectively, to (k¯1 ξε (t), k¯1 ηε (t)) and (k¯1 ξ˜ε (t), k¯1 η˜ε (t)). In the similar way, we index random variables, in particular hitting times, phase spaces and other related sets, conditions, probabilities, expectations, and other quantities and objects related to the multi-alternating regenerative processes (k¯ n ξε (t), k¯ n ηε (t)) and (k¯ n ξ˜ε (t), k¯ n η˜ε (t)) by the left lower index k¯ n (in such a way as k¯ n f ), in order to distinguish these random variables, phase spaces and other sets, probabilities, expectations, and other quantities and objects, for n = 1, . . . , m − 1. To obtain similar forms for recurrent relations connecting the above random variables, probabilities and expectations, and other quantities and objects obtained by excluding the sequences of states k¯ n−1 and k¯ n from the phase space X, we also use the left lower index k¯0 (for the “empty” sequence k¯ 0 = ), for n = 1. Thus, we change the notation from (ξε (t), ηε (t)) and (ξ˜ε (t), η˜ε (t)), respectively, to (k¯0 ξε (t), k¯0 ηε (t)) and (k¯0 ξ˜ε (t), k¯0 η˜ε (t)). In the similar way, we index k¯0 X = X and other related sets, conditions, probabilities, expectations, and other quantities and objects related to the multi-alternating regenerative processes (k¯0 ξε (t), k¯0 ηε (t)) = (ξε (t), ηε (t)) and (k¯0 ξ˜ε (t), k¯0 η˜ε (t)). Let us describe the recurrent algorithm described above in more detail. 11.2.1.2 The First Step of Recurrent Algorithm of Time–Space Aggregation of Regeneration Times Based on Total Removing of Virtual Transitions. The initial conditions k¯ 0 G1 = G1 and k¯ 0 H1 = H1 are assumed to be satisfied for the multi-alternating regenerative processes (k¯0 ξε (t), k¯0 ηε (t)) = (ξε (t), ηε (t)). The first step of the phase space reduction algorithm includes two sub-steps. At the first sub-step, the multi-alternating regenerative process (ξε (t), ηε (t)) = (k¯0 ξε (t), k¯0 ηε (t)) is transformed into the multi-alternating regenerative process (k¯0 ξ˜ε (t), k¯0 η˜ε (t)) by applying the procedure of total removing virtual transitions (described in Sects. 10.2.1 and 11.1.1) to the modulating semi-Markov process k¯0 ηε (t). The resulting modulating semi-Markov process k¯0 η˜ε (t) has the phase space k¯0 X = X. The first regenerative component k¯0 ξε (t) = k¯1 ξ˜ε (t) does not change. At the second sub-step, a state k1 , such that the condition k¯ 1 M = k1 M is satisfied, should be chosen. Then, the multi-alternating regenerative process (k¯0 ξ˜ε (t), k¯0 η˜ε (t))

260

11 Time-space aggregation of regeneration times

is transformed into the multi-alternating regenerative process (k¯1 ξε (t), k¯1 ηε (t)), by applying the procedure of one-state reduction of phase space (described in Sects. 10.3.1 and 11.1.2) to the modulating semi-Markov process k¯0 η˜ε (t). The resulting reduced modulating semi-Markov process k¯1 ηε (t) has the phase space k¯1 X. The first regenerative component k¯1 ξ˜ε (t) = k¯1 ξε (t) does not change. In the above procedure, it is initially assumed that the conditions k¯ 0 G1 = G1 , ¯k0 H1 = H1 , k¯ 0 I1 = I1 , k¯ 0 IH,1 = IH,1 , k¯ 0 L1 = L1 , k¯ 0 LH,1 = LH,1 , k¯ 0 J1 = J1 , k¯ 0 K1 = K1 , k¯ 0 R = R are satisfied for the multi-alternating regenerative processes (k¯0 ξε (t), k¯0 ηε (t)) = (ξε (t), ηε (t)). In addition, it is necessary to select a state k1 so that the condition k¯ 1 M = k1 M is satisfied. This makes it possible to implement the second sub-step of the above algorithm and transform the multi-alternating regenerative process (k¯0 ξ˜ε (t), k¯0 η˜ε (t)) into the multi-alternating regenerative process (k¯1 ξε (t), k¯1 ηε (t)). From Lemmas 10.4–10.10 it follows that the conditions k¯ 0 G1 , k¯ 0 H1 , k¯ 0 I1 , k¯ 0 IH,1 , k¯ 0 L1 , k¯ 0 LH,1 , k¯ 0 J1 , k¯ 0 K1 , and k¯ 0 R are satisfied for the multi-alternating regenerative ˜ , k¯ H ˜ , k¯ I˜ , k¯ I˜ , k¯ L ˜ , processes (k¯ ξ˜ε (t), k¯ η˜ε (t)) in the form of conditions k¯ G 0

0

0

1

0

1

0

1

0

H,1

0

1

˜ ˜ ˜ ˜ k¯ 0 LH,1 , k¯ 0 J1 , k¯ 0 K1 , and k¯ 0 R. Also, from Lemmas 10.19–10.24 it follows that the conditions k¯0 G1 , k¯ 0 H1 , k¯ 0 I1 , k¯ 0 IH,1 , k¯ 0 L1 k¯ 0 LH,1 , k¯0 J1 , k¯0 K1 , k¯0 R hold for the multi-alternating regenerative processes (k¯1 ξε (t), k¯1 ηε (t)) in the form of conditions k¯ 1 G1 , k¯ 1 H1 , k¯ 1 I1 , k¯ 1 IH,1 , k¯ 1 L1 , k¯ 1 LH,1 , k¯ 1 J1 , k¯ 1 K1 , k¯ 1 R. 11.2.1.3 The Recurrent Algorithm of Time–Space Aggregation of Regeneration Times Based on Total Removing of Virtual Transitions. The first step of phase space reduction algorithm described in Sect. 11.2.1.2 can be recurrently repeated n times, for some 1 ≤ n ≤ m − 1. The n-th step of the phase space reduction algorithm includes two sub-steps similar to those described above for the first step. The only difference is that the corresponding sub-algorithms must be applied to the multi-alternating regenerative process (k¯ n−1 ξε (t), k¯ n−1 ηε (t)) instead of the multi-alternating regenerative process (k¯0 ξε (t), k¯0 ηε (t)). At the first sub-step, the multi-alternating regenerative process (k¯ n−1 ξε (t), k¯ n−1 ηε (t)) is transformed into the multi-alternating regenerative process (k¯ n−1 ξ˜ε (t), k¯ n−1 η˜ε (t)) by applying the procedure of total removal virtual transitions (described in Sects. 10.2.1 and 11.1.1) to the modulating semi-Markov process k¯ n−1 ηε (t). The resulting modulating semi-Markov process k¯ n−1 η˜ε (t) has the phase space k¯ n−1 X = X\{k1, . . . , k n−1 }. The first regenerative component k¯ n−1 ξε (t) = k¯ n−1 ξ˜ε (t) does not change. At the second sub-step, the multi-alternating regenerative process (k¯ n−1 ξ˜ε (t), k¯ n−1 η˜ε (t)) is transformed into the multi-alternating regenerative process (k¯ n ξε (t), k¯ n ηε (t)) by applying the procedure of one-state reduction of phase space (described in Sects. 10.3.1 and 11.1.2) to the modulating semi-Markov process k¯ n−1 η˜ε (t). The resulting modulating semi-Markov process k¯ n ηε (t) has the reduced phase space k¯ n X = kn−1 X \ {k n } = X \ {k1, . . . , k n }. The first regenerative component ˜ k¯ n−1 ξε (t) = k¯ n ξε (t) does not change.

11.2 Aggregation of regeneration times based on total removing of virtual transitions

261

The conditions k¯ n−1 G1 , k¯ n−1 H1 , k¯ n−1 I1 , k¯ n−1 IH,1 , k¯ n−1 L1 , k¯ n−1 LH,1 , k¯ n−1 J1 , k¯ n−1 K1 , k¯ n−1 R are satisfied for the multi-alternating regenerative processes (k¯ n−1 ξε (t), k¯ n−1 ηε (t)). This is the result of the (n − 1)-th step of the above recurrent algorithm. To implement n−1 steps in the above algorithm, it is also necessary to assume that the condition k¯ n−1 M (the condition kl M is satisfied for the semi-Markov processes k¯l−1 η˜ε (t), for l = 1, . . . , n − 1) is satisfied. At the n-th step, this assumption should be extended to the assumption that the condition k¯ n M is satisfied. This makes it possible to implement the second sub-algorithm of transforming the multi-alternating regenerative process (k¯ n−1 ξ˜ε (t), k¯ n−1 η˜ε (t)) into the multi-alternating regenerative process (k¯ n ξε (t), k¯ n ηε (t)). From Lemmas 10.4–10.10 it follows that the conditions k¯ n−1 G1 , k¯ n−1 H1 , k¯ n−1 I1 , ¯kn−1 IH,1 , k¯ n−1 L1 , k¯ n−1 LH,1 , k¯ n−1 J1 , k¯ n−1 K1 , k¯ n−1 R are satisfied for the multi-alternating ˜ , k¯ H ˜ , regenerative processes (k¯ n−1 ξ˜ε (t), k¯ n−1 η˜ε (t)) in the form of conditions k¯ n−1 G 1 n−1 1 ˜ , k¯ L ˜ ˜ , k¯ R. ˜ , k¯ J˜ , k¯ K I˜ , k¯ I˜ , k¯ L k¯ n−1

1

n−1

H,1

n−1

1

n−1

H,1

1

n−1

n−1

1

n−1

From Lemmas 10.19–10.24 it follows that the conditions k¯ n−1 G1 , k¯ n−1 H1 , k¯ n−1 I1 , k¯ n−1 IH,1 , k¯ n−1 L1 , k¯ n−1 LH,1 , k¯ n−1 J1 , k¯ n−1 K1 , k¯ n−1 R are satisfied for the multi-alternating regenerative processes (k¯ n ξε (t), k¯ n ηε (t)) in the form of conditions k¯ n G1 , k¯ n H1 , k¯ n I1 , k¯ n IH,1 , k¯ n L1 , k¯ n LH,1 , k¯ n J1 , k¯ n K1 , k¯ n R. The following recurrent relations connect, for every ε ∈ [0, 1], the transition probabilities of the embedded Markov chains k¯ n−1 ηε,r , k¯ n−1 η˜ε,r , and k¯ n ηε,r , for n = 1, . . . , m − 1, k¯ n−1 p˜ε,i j

= I( j  i)

k¯ n−1 pε,i j k¯ n−1 p¯ε,ii

k¯ n pε,i j

=

k¯0 pε,i j

= pε,i j , for i, j ∈

k¯ n−1 p˜ε,i j

+

, for i, j ∈

k¯ n−1 X,

k¯ n−1 p˜ε,ik n k¯ n−1 p˜ε,k n j ,

for i, j ∈

k¯ n X,

(11.8)

where k¯0 X

= X.

(11.9)

Also, the following recurrent relations connect, for every ε ∈ [0, 1], the normalisation functions k¯ n−1 u˜ ·,i , used for the multi-alternating regenerative processes (k¯ n−1 ξ˜ε (t), k¯ n−1 η˜ε (t)), and the normalisation functions k¯ n u ·,i , used for multi-alternating regenerative processes (k¯ n ξε (t), k¯ n ηε (t)), for n = 1, . . . , m − 1, k¯ n−1 u˜ ε,i

=

−1 k¯ n−1 p¯ε,ii k¯ n−1 uε,i,

k¯ n uε,i

=

k¯ n−1 u˜ ε,i,

for i ∈

for i ∈

k¯ n X,

k¯ n−1 X,

(11.10)

where k¯0 uε,i

= uε,i, for i ∈

k0 X

= X.

(11.11)

The relation (11.10) implies that the following formulas take place for the above normalisation functions, for n = 1, . . . , m − 1,

262

11 Time-space aggregation of regeneration times

k¯ n−1 u˜ ε,i

=

n−1 

−1 k¯r p¯ε,ii

uε,i, for i ∈

k¯ n−1 X,

r=0 k¯ n uε,i

=

k¯ n−1 u˜ ε,i ,

for i ∈

k¯ n X.

(11.12)

It is worth noting that the conditions IH,1 and LH,1 imply that the normalisation functions k¯ n−1 u˜ ·,i, i ∈ k¯ n−1 X and k¯ n uε,i, i ∈ k¯ n X belong to the family of asymptotically comparable functions H. The above-mentioned condition k¯ n M takes the following form, for n = 1, . . ., m − 1: k¯ n M: k¯l−1 w0,kl ,i

= limε→0

˜ ε, k l k¯ l−1 u ˜ ε, i k¯ l−1 u

∈ [0, ∞), i ∈

k¯l−1 X,

for l = 1, . . . , n.

This condition means that at each step of successive exclusion of the states k1, . . . , k n from the phase space X, the state kl is selected from the set of least absorbing states k¯l−1 W0 ⊆ k¯l−1 X for the semi-Markov process k¯l−1 η˜ε (t), sequentially, for l = 1, . . . , n. The conditions G1 , H1 , IH,1 , and LH,1 imply that limits in the asymptotic relations given in the condition k¯ n M exist and can be recurrently calculated using Lemmas 8.21 –8.91 . We denote by Mn the set of sequences of states k¯ n , such that all asymptotic relations appearing in the condition k¯ n M hold (with the limits indicated in this condition). According to Lemmas 9.201 and 9.231 , the conditions G1 , H1 , IH,1 , and LH,1 imply that the set Mn is not empty. Thus, the condition k¯ n M, in fact, means only that the sequence k¯ n is selected from the set Mn .

11.2.2 Summary of Algorithm of Time–Space Aggregation for Regeneration Times Based on Total Removing of Virtual Transitions This algorithm is realised in sequential recurrent steps of constructing the sequence of multi-alternating regenerative processes with totally removed virtual transitions and reduced phase spaces for the corresponding modulating semi-Markov processes, for some 1 ≤ n ≤ m − 1. This sequence of recurrent transformations can be represented by the following symbolic diagram: (ξε (t), ηε (t)) = (k¯0 ξε (t), k¯0 ηε (t)) → (k¯0 ξ˜ε (t), k¯0 η˜ε (t)) → (k¯1 ξε (t),

k¯1 ηε (t))

··· → (k¯ n−1 ξ˜ε (t),

k¯ n−1 η˜ε (t))

→ (k¯ n ξε (t),

k¯ n ηε (t)).

(11.13)

11.2 Aggregation of regeneration times based on total removing of virtual transitions

263

The corresponding recurrent formulas for finding transition characteristics of the above perturbed multi-alternating regenerative process are given in Sects. 10.2.1 and 10.3.1. The initial regularity conditions G1 , H1 and the perturbation conditions I1 , IH,1 , L1 , LH,1 , J1 , K1 , R are imposed on the original multi-alternating regenerative processes (ξε (t), ηε (t)). In addition, the condition k¯ n M should be assumed to be satisfied before implementing the second sub-step of n-th step of the above algorithm, for n = 1, . . . , m − 1. The asymptotic recurrent algorithms presented in Sects. 10.2.1, 10.3.1 and Lemmas 10.4–10.10, 10.19–10.24 ensure the fulfilment of similar sets of conditions for the corresponding transformed multivariate regenerative processes. This can be represented in the following symbol diagram: G1, H1, I1, IH,1, L1, LH,1, J1, K1, R

= k¯ 0 G1, k¯ 0 H1, k¯ 0 I1, k¯ 0 IH,1, k¯ 0 L1, k¯ 0 LH,1, k¯ 0 J1, k¯ 0 K1, k¯ 0 R

⇓ ˜ ˜ ˜ ˜ k¯ 0 G1, k¯ 0 H1, k¯ 0 I1, k¯ 0 IH,1,

k¯ 0

˜ , L 1

k¯ 0

˜ ˜ , R

˜ L , J˜ , K H,1 k¯ 0 1 k¯ 0 1 k¯ 0

⇓ k¯ 1 M

⇒ k¯ 1 G1,

k¯ 1 H1, k¯ 1 I1, k¯ 1 IH,1, k¯ 1 L1, k¯ 1 LH,1, k¯ 1 J1, k¯ 1 K1, k¯ 1 R

⇓ ··· ⇓ k¯ n−1 M ⇒ k¯ n−1 G1,

k¯ n−1 H1, k¯ n−1 I1, k¯ n−1 IH,1, k¯ n−1 L1, k¯ n−1 LH,1, k¯ n−1 J1,

⇓ ˜ ˜ ˜ ˜ , k¯ L ˜ k¯ n−1 G1, k¯ n−1 H1, k¯ n−1 I1, k¯ n−1 I˜ H,1, k¯ n−1 L J˜ , , 1 n−1 H,1 k¯ n−1 1

k¯ n−1 K1, k¯ n−1 R .

˜ ˜ k¯ n−1 K1, k¯ n−1 R

⇓ k¯ n M

⇒ k¯ n G1,

k¯ n H1, k¯ n I1, k¯ n IH,1, k¯ n L1, k¯ n LH,1, k¯ n J1, k¯ n K1, k¯ n R .

(11.14)

The corresponding recurrent relations for “re-calculating” the above sets of conditions are given in Sects. 10.2.1 and 10.3.1. Importantly, the sequence k¯ n = k1, . . . , k n must satisfy the condition k¯ n M. The following theorem summarises the above remarks. Theorem 11.1 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Then: (i) If the condition k¯ n−1 M is additionally satisfied for some 1 ≤ n ≤ m − 1, then the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R are satisfied for the multialternating regenerative processes (k¯ n−1 ξ˜ε (t), k¯ n−1 η˜ε (t)), in the form of conditions, ˜ , k¯ H ˜ , k¯ I˜ , k¯ I˜ , k¯ L ˜ , k¯ L ˜ ˜ , k¯ R. ˜ G , k¯ J˜ , k¯ K k¯ n−1

1

n−1

1

n−1

1

n−1

H,1

n−1

1

n−1

H,1

n−1

1

n−1

1

n−1

264

11 Time-space aggregation of regeneration times

(ii) The following relation takes place, for i ∈

k¯ n−1 X,

Pi {k¯ n−1 ξ˜ε (t) = ξε (t), t ≥ 0} = 1.

(11.15)

(iii) If the condition k¯ n M is additionally satisfied for some 1 ≤ n ≤ m − 1, then the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , and R are satisfied for the multialternating regenerative processes (k¯ n ξε (t), k¯ n ηε (t)) in the form of conditions k¯ n G1 , k¯ n H1 , k¯ n I1 , k¯ n IH,1 , k¯ n L1 , k¯ n LH,1 , k¯ n J1 , k¯ n K1 , k¯ n R. (iv) The following relation takes place, for i ∈

k¯ n X,

Pi {k¯ n ξε (t) = ξε (t), t ≥ 0} = 1.

(11.16)

Proof This theorem is a direct corollary of Theorem 9.31 and Lemmas 10.4– 10.10, 10.19 - 10.24. 

11.3 Time–Space Aggregation for Regeneration Times Based on Partial Removing of Virtual Transitions This section describes an asymptotic recurrent algorithm for space-time aggregation for regeneration times for multivariate regenerative processes, based on partial removal of virtual transitions from the trajectories of modulating semi-Markov processes and reduction of its phase space.

11.3.1 Algorithm of Time–Space Aggregation for Regeneration Times Based on Partial Removing of Virtual Transitions 11.3.1.1 Notation. This algorithm is based on the recurrent application of the procedure of partial removal of virtual transitions and the corresponding procedure of exclusion of one state from the phase space of modulating semi-Markov processes. These procedures are described in Sects. 10.2.2, 10.3.2, and 11.1. The notation used in this version of the algorithm is similar to the notation presented in Sect. 11.2.1.1. The key element of this algorithm is again the procedure of successive exclusion of a sequence of states k¯ n = k1, k2, . . . , k n , for some 1 ≤ n ≤ m − 1, from the phase space X of the modulating semi-Markov processes ηε (t). This procedure reduces the phase space X to the space k¯ n X = X \ {k1, k2, . . . , k n }. We use the abbreviated notation (k¯ n ξε (t), k¯ n ηε (t)) for the multi-alternating regenerative process obtained by successive exclusion of the sequence of states k¯ n = k1, . . . , k n from the phase space X of the modulating semi-Markov process ηε (t).

11.3 Aggregation of regeneration times based on partial removing of virtual transitions

265

Process (k¯ n ξε (t), k¯ n ηε (t)) is the result of the n-th step of the algorithm that includes two sub-steps. At the first sub-step, virtual transitions of the form i → i are partially (for i ∈ k¯ n−1 X ∩ K[k n ]) removed from trajectories of the modulating semi-Markov process k¯ n−1 ηε (t) for the multi-alternating regenerative process (k¯ n−1 ξε (t), k¯ n−1 ηε (t)). We use the abbreviated notation (k¯ n ξ˜ε (t), k¯ n η˜ε (t)) for the obtained multi-alternating regenerative process. At the second sub-step, state k n is excluded from the phase space of the modulating semi-Markov process k¯ n η˜ε (t), resulting in the multi-alternating regenerative process (k¯ n ξε (t), k¯ n ηε (t)). Note that the multi-alternating regenerative process (k¯ n ξ˜ε (t), k¯ n η˜ε (t)) has the phase space Z × k¯ n−1 X, while the multi-alternating regenerative process (k¯ n ξε (t),  k¯ n ηε (t)) has the phase space Z × k¯ n X. In particular, we change the notation from (k1 ξ˜ε (t), k1 η˜ε (t)) and (k1 ξε (t), k1 ηε (t)) to, respectively, (k¯1 ξ˜ε (t), k¯1 η˜ε (t)) and (k¯1 ξε (t), k¯1 ηε (t)). In the similar way, we index random variables, in particular hitting times, phase spaces and other related sets, conditions, probabilities, expectations, and other quantities and objects related to the multi-alternating regenerative processes (k¯ n ξ˜ε (t), k¯ n η˜ε (t)) and (k¯ n ξε (t), k¯ n ηε (t)) by the left lower index k¯ n (in such a way as k¯ n f ), in order to distinguish these random variables, phase spaces and other sets, probabilities, expectations, and other quantities and objects, for n = 1, . . . , m − 1. To obtain similar forms for recurrent relations connecting the above random variables, probabilities and expectations and other quantities and objects obtained by excluding the sequences of states k¯ n−1 and k¯ n from the phase space, we also use the left lower index k¯0 (for the “empty” sequence k¯ 0 = ), for n = 1. Thus, we change the notation from (ξε (t), ηε (t)) to (k¯0 ξε (t), k¯0 ηε (t)). In the similar way, we index k¯0 X = X and other related sets, conditions, probabilities, expectations, and other quantities and objects related to the multi-alternating regenerative process (k¯0 ξε (t), k¯0 ηε (t)) = (ξε (t), ηε (t)). Let us describe in more detail the above-mentioned recurrent algorithm. 11.3.1.2 The First Step of Recurrent Algorithm of Time–Space Aggregation of Regeneration Times Based on Partial Removing of Virtual Transitions. The initial conditions k¯ 0 G1 = G1 and k¯ 0 H1 = H1 should be assumed to hold for the multi-alternating regenerative process (k¯0 ξε (t), k¯0 ηε (t)) = (ξε (t), ηε (t)). The first step of the phase space reduction algorithm includes two sub-steps. At the first sub-step, a state k1 , such that condition k¯ 1 M = k1 M is satisfied, must be selected. Then, the multi-alternating regenerative process (ξε (t), ηε (t)) = (k¯0 ξε (t), k¯0 ηε (t)) is transformed into the multi-alternating regenerative process   (k¯1 ξ˜ε (t), k¯1 η˜ε (t)) = (ξ˜ε,K[k (t), η˜ε,K[k (t)), by applying the procedure of partial re1] 1] moving virtual transitions i → i (for i ∈ K[k1 ] = k¯0 K[k1 ]) (described in Sect. 10.2.2 and 11.1.1) to the modulating semi-Markov process k¯0 ηε (t). The resulting modulating semi-Markov process k¯1 η˜ε (t) has the phase space k¯0 X = X. The first regenerative component k¯0 ξε (t) = k¯1 ξ˜ε (t) does not change.

266

11 Time-space aggregation of regeneration times

At the second sub-step, the multi-alternating regenerative process (k¯1 ξ˜ε (t), k¯1 η˜ε (t)) is transformed into the multi-alternating regenerative process (k¯1 ξε (t), k¯1 ηε (t)), by applying the procedure of one-state reduction of phase space (described in Sect. 10.3.2 and 11.1.2) to the modulating semi-Markov process k¯1 η˜ε (t). The resulting reduced modulating semi-Markov process k¯1 ηε (t) has the phase space k¯1 X. The first regenerative component k¯1 ξ˜ε (t) = k¯1 ξε (t) does not change. In the above procedure, it is initially assumed that k¯0 G1 = G1 , k¯0 H1 = H1 ,        k¯0 I1 = I1 , k¯ 0 IH,1 = IH,1 , k¯0 L1 = L1 , k¯ 0 LH,1 = LH,1 , k¯ 0 J1 = J1 , k¯ 0 K1 = K1 , k¯ 0 R = R are satisfied for the multi-alternating regenerative processes (k¯0 ξε (t), k¯0 ηε (t)) = (ξε (t), ηε (t)). In addition, it is necessary to select a state k1 so that the condition k¯ 1 M = k1 M is satisfied. This makes it possible to implement both sub-steps of the algorithm and transform the multi-alternating regenerative process (k¯0 ξε (t), k¯0 ηε (t)) into the multialternating regenerative process (k¯1 ξ˜ε (t), k¯1 η˜ε (t)) and, then, the multi-alternating regenerative process (k¯1 ξ˜ε (t), k¯1 η˜ε (t)) into the multi-alternating regenerative process (k¯1 ξε (t), k¯1 ηε (t)).  , From Lemmas 10.11–10.18 it follows that the conditions k¯0 G1 , k¯ 0 H1 , k¯ 0 I1 , k¯ 0 IH,1      k¯ 0 L1 , k¯ 0 LH,1 , k¯ 0 J1 , k¯ 0 K1 , k¯ 0 R are satisfied for the multi-alternating regenerative ˜  , k¯ H ˜  , k¯ I˜  , k¯ I˜  , k¯ L ˜ , processes (k¯ ξ˜ε (t), k¯ η˜ε (t)) in the form of conditions k¯ G 1

1

1

1

1

1

1

1

1

H,1

1

1

˜ ˜ ˜ ˜ k¯ 1 LH,1 , k¯ 1 J1 , k¯ 1 K1 , k¯ 1 R . Also, from Lemmas 10.26–10.31 it follows that the conditions k¯0 G1 , k¯ 0 H1 , k¯ 0 I1 ,       k¯ 0 IH,1 , k¯ 0 L1 k¯ 0 LH,1 , k¯0 J1 , k¯0 K1 , k¯0 R are satisfied for the multi-alternating regener   ative processes (k¯1 ξε (t), k¯1 ηε (t)) in the form of conditions k¯ 1 G1 , k¯ 1 H1 , k¯ 1 I1 , k¯ 1 IH,1 ,      k¯ 1 L1 , k¯ 1 LH,1 , k¯ 1 J1 , k¯ 1 K1 , and k¯ 1 R1 . 11.3.1.3 The Recurrent Algorithm of Time–Space Aggregation of Regeneration Times Based on Partial Removing of Virtual Transitions. The first step of phase space reduction algorithm described in Sect. 11.3.1.3 can be recurrently repeated n times, for any 1 ≤ n ≤ m − 1. The n-th step of the phase space reduction algorithm includes two sub-steps similar to those described above for the first step. The only difference is that the corresponding sub-algorithms must be applied to the multi-alternating regenerative process (k¯ n−1 ξε (t), k¯ n−1 ηε (t)), instead of the multi-alternating regenerative process (k¯0 ξε (t), k¯0 ηε (t)). At the first sub-step, the multi-alternating regenerative process (k¯ n−1 ξε (t),   ˜ k¯ n−1 ηε (t)) is transformed into the semi-Markov process (k¯ n ξε (t), k¯ n η˜ε (t)), by applying the procedure of partial removal virtual transitions (described in Sects. 10.2.2 and 11.1.1) to the modulating semi-Markov process k¯ n−1 ηε (t). The resulting modulating semi-Markov process k¯ n η˜ε (t) has the phase space k¯ n−1 X = X \ {k1, . . . , k n−1 }. The first regenerative component k¯ n−1 ξε (t) = k¯ n ξ˜ε (t) does not change. At the second sub-step, the multi-alternating regenerative process (k¯ n ξ˜ε (t),   k¯ n η˜ε (t)) is transformed into the multi-alternating regenerative process (k¯ n ξε (t),  k¯ n ηε (t)), by applying the procedure of one-state reduction of phase space (described in Sects. 10.3.2 and 11.1.2) to the modulating semi-Markov process k¯ n−1 η˜ε (t)).

11.3 Aggregation of regeneration times based on partial removing of virtual transitions

267

The resulting modulating semi-Markov process k¯ n ηε (t) has the reduced phase space k¯ n X = kn−1 X \ {k n } = X \ {k1, . . . , k n }. The first regenerative component  ˜ k¯ n ξε (t) = k¯ n ξε (t) does not change.   , k¯ n−1 L1 , k¯ n−1 LH,1 , k¯ n−1 J1 , k¯ n−1 K1 , The conditions k¯ n−1 G1 , k¯ n−1 H1 , k¯ n−1 I1 , k¯ n−1 IH,1 and k¯ n−1 R are satisfied for the multi-alternating regenerative processes (k¯ n−1 ξε (t),  k¯ n−1 ηε (t)). This is the result of the (n − 1)-th step of the above recurrent algorithm. To implement n − 1 steps in the above algorithm, it is also necessary to assume that the condition k¯ n−1 M (the condition kl M is satisfied for the semi-Markov processes k¯l−1 ηε (t), for l = 1, . . . , n − 1). At the n-th step, this assumption should be extended to the assumption that the condition k¯ n M is satisfied. This makes it possible to implement the n-th step of the algorithm transforming the multi-alternating regenerative process (k¯ n−1 ξε (t), k¯ n−1 ηε (t)) into the multi-alternating regenerative process (k¯ n ξ˜ε (t), k¯ n η˜ε (t)), and, then, into the multi-alternating regenerative process (k¯ n ξε (t), k¯ n ηε (t)). From Lemmas 10.11–10.18 it follows that the conditions k¯ n−1 G1 , k¯ n−1 H1 , k¯ n−1 I1 , ¯kn−1 IH,1 , k¯ n−1 L1 , k¯ n−1 LH,1 , k¯ n−1 J1 , k¯ n−1 K1 , k¯ n−1 R are satisfied for the multi-alternating ˜  , k¯ H ˜  , I˜  , regenerative processes (k¯ n ξ˜ε (t), k¯ n η˜ε (t)) in the form of conditions k¯ n G 1 n 1 k¯ n 1 ˜  , k¯ L ˜  , k¯ L ˜  , k¯ J˜  , k¯ K ˜  , and k¯ R ˜ . k¯ I n

H,1

n

1

n

H,1

n

1

n

1

n

From Lemmas 10.26–10.31 it follows that the conditions k¯ n−1 G1 , k¯ n−1 H1 , k¯ n−1 I1 ,       k¯ n−1 IH,1 , k¯ n−1 L1 , k¯ n−1 LH,1 , k¯ n−1 J1 , k¯ n−1 K1 , k¯ n−1 R are satisfied for the multi-alternating   regenerative processes (k¯ n ξε (t), k¯ n ηε (t)) in the form of conditions k¯ n G1 , k¯ n H1 , k¯ n I1 ,       k¯ n IH,1 , k¯ n L1 , k¯ n LH,1 , k¯ n J1 , k¯ n K1 , and k¯ n R . The following recurrent relations connect, for every ε ∈ (0, 1], the transition   ,  probabilities of the embedded Markov chains k¯ n−1 ηε,r and k¯ n η˜ε,r k¯ n ηε,r , for n = 1, . . . , m − 1,  for i ∈ k¯ n−1 X \ k¯ n−1 K[k n ], j ∈ k¯ n−1 X, k¯ n−1 pε,i j   p ˜ = k¯ n−1 p ε, i j k¯ n ε,i j I( j  i) ¯ p¯  for i ∈ k¯ n−1 X ∩ k¯ n−1 K[k n ], j ∈ k¯ n−1 X, k n−1

 k¯ n pε,i j

where

ε, ii

 ⎧ ⎪ k¯ n p˜ε,i j ⎪ ⎪ ⎨ ⎪ for i ∈ k¯ n X \ k¯ n−1 K[k n ], j ∈ k¯ n X, =    p ˜ ⎪ k¯ n ε,i j + k¯ n p˜ε,ik n k¯ n p˜ε,k n j ⎪ ⎪ ⎪ for i ∈ k¯ n X ∩ k¯ n−1 K[k n ], j ∈ k¯ n X, ⎩

 k¯0 pε,i j

= pε,i j , for i, j ∈

and k¯ n−1 K[k n ]

= {i ∈

k¯ n X

:

k¯0 X

= X,

 k¯ n−1 p1,ik n

> 0} ∪ {k n }.

(11.17)

(11.18) (11.19)

Also, the following recurrent relations connect, for every ε ∈ [0, 1], the normalisation functions k¯ n u˜ ·,i , used for the multi-alternating regenerative processes (k¯ n ξ˜ε (t), k¯ n η˜ε (t)), and the normalisation functions k¯ n u ·,i , used for the multi-

268

11 Time-space aggregation of regeneration times

alternating regenerative processes (k¯ n ξε (t),  k¯ n u˜ ε,i =

where



 k¯ n−1 uε,i −1  k¯ n−1 p¯ε,ii k¯ n−1 uε,i

 k¯ n uε,i

=

 k¯ n u˜ ε,i,

 k¯0 uε,i

= uε,i, for i ∈

for i ∈

 k¯ n ηε (t)),

for i ∈ for i ∈

for n = 1, . . . , m − 1,

k¯ n−1 X k¯ n−1 X

\ k¯ n−1 K[k n ], ∩ k¯ n−1 K[k n ],

k¯ n X,

k¯0 X

(11.20)

= X.

(11.21)

  noting that, by the conditions IH,1 and LH,1 , the normalisation func ∈ k¯ n−1 X, n = 1, . . . , m and k¯ n uε,i, i ∈ k¯ n X belong to the family of

It is worth tions k¯ n u˜ ·,i, i asymptotically comparable functions H. The above-mentioned condition k¯ n M takes the following form, for n = 1, . . ., m − 1: k¯ n M



 : k¯l−1 w0,k = limε→0 l ,i

 ˜ ε, k¯ l u kl  u ˜ ¯ k l ε, i

∈ [0, ∞), i ∈

k¯l−1 K[k l ],

for l = 1, . . . , n.

11.3.2 Summary of Algorithm of Time–Space Aggregation of Regeneration Times Based on Partial Removing of Virtual Transitions This algorithm is realised in sequential recurrent steps of constructing the sequence of multi-alternating regenerative processes with partially removed virtual transitions and reduced phase spaces for the corresponding modulating semi-Markov processes, for some 1 ≤ n ≤ m − 1. This sequence of recurrent transformations can be represented by the following symbolic diagram: (ξε (t), ηε (t)) = (k¯0 ξε (t), k¯0 ηε (t)) → (k¯1 ξ˜ε (t), k¯1 η˜ε (t)) → (k¯1 ξε (t),

 k¯1 ηε (t))

··· → (k¯ n ξ˜ε (t),

 k¯ n η˜ε (t))

→ (k¯ n ξε (t),

 k¯ n ηε (t)).

(11.22)

The corresponding recurrent formulas for finding transition characteristics of the above perturbed multi-alternating regenerative process are given in Sects. 10.2.2 and 10.3.2. The initial regularity conditions G1 , H1 and the perturbation conditions I1 , IH,1 , L1 , LH,1 , J1 , K1 , R are imposed on the original multi-alternating regenerative process (ξε (t), ηε (t)). In addition, the condition k¯ l M should be assumed to be satisfied before the implementing the l-th step of the algorithm, for l = 1, . . . , n. The asymptotic recurrent algorithms presented in Sects. 10.2.2, 10.3.2 and Lemmas 10.11–10.18, 10.26–10.31 ensure the fulfilment of similar sets of conditions for

11.3 Aggregation of regeneration times based on partial removing of virtual transitions

269

the corresponding transformed multi-alternating regenerative processes. This can be represented in the following symbolic diagram: G1, H1, I1, IH,1, L1, LH,1, J1, K1, R

= k¯ 0 G1, k¯ 0 H1, k¯ 0 I1, k¯ 0 IH,1, k¯ 0 L1, k¯ 0 LH,1, k¯ 0 J1, k¯ 0 K1, k¯ 0 R

⇓ ˜ ˜  ˜ ˜ k¯ 1 M ⇒ k¯ 1 G1, k¯ 1 H1, k¯ 1 I1, k¯ 1 IH,1, 

k¯ 1

˜ , L 1

k¯ 1

˜  , k¯ J˜  , k¯ K ˜ , R ˜ 

L H,1 1 1 1 1 k¯ 1

⇓ k¯ 1 G1,

k¯ 1 H1, k¯ 1 I1, k¯ 1 IH,1, k¯ 1 L1, k¯ 1 LH,1, k¯ 1 J1, k¯ 1 K1, k¯ 1 R

⇓ ··· ⇓ 

k¯ n M

⇒

˜  , k¯ H ˜  , I˜ , I˜ , k¯ n G 1 n 1 k¯ n 1 k¯ n H,1 k¯ n

˜  , k¯ L ˜  , J˜  , L 1 n H,1 k¯ n 1

˜ ˜ k¯ n K1, k¯ n R

⇓ k¯ n G1,

        k¯ n H1, k¯ n I1, k¯ n IH,1, k¯ n L1, k¯ n LH,1, k¯ n J1, k¯ n K1, k¯ n R .

(11.23)

The corresponding recurrent relations for “re-calculating” the above sets of conditions are given in Sects. 10.2.2 and 10.3.2. Importantly, the sequence k¯ n = k1, . . . , k n must satisfy condition k¯ n M1 . The following theorem summarises the above remarks. Theorem 11.2 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)), and the condition k¯ n M is satisfied. Then: (i) The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R are satisfied for the multialternating regenerative processes (k¯ n ξ˜ε (t), k¯ n η˜ε (t)), in the forms of conditions, ˜ ˜  ˜ ˜ ˜ ˜ ˜ ˜ ˜ k¯ n G1 , k¯ n H1 , k¯ n I1 , k¯ n IH,1 , k¯ n L1 , k¯ n LH,1 , k¯ n J1 , k¯ n K1 , k¯ n R . (ii) The following relation takes place, for i ∈

k¯ n−1 X,

Pi {k¯ n ξ˜ε (t) = ξε (t), t ≥ 0} = 1.

(11.24)

(iii) The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , and R are satisfied for the multi-alternating regenerative processes (k¯ n ξε (t), k¯ n ηε (t)) in the form of conditions          k¯ n G1 , k¯ n H1 , k¯ n I1 , k¯ n IH,1 , k¯ n L1 , k¯ n LH,1 , k¯ n J1 , k¯ n K1 , k¯ n R . (iv) The following relation takes place, for i ∈

k¯ n X,

Pi {k¯ n ξε (t) = ξε (t), t ≥ 0} = 1.

(11.25)

Proof This theorem is a direct corollary of Lemmas 10.11–10.18, 10.26–10.31. 

270

11 Time-space aggregation of regeneration times

11.4 Comparison of Recurrent Algorithms of Time–Space Aggregation for Regeneration Times This section provides a comparative analysis of asymptotic recurrent algorithms (based, respectively, on total or partial removal of virtual transitions) of time–space aggregation for regeneration times of perturbed multi-alternating regenerative processes.

11.4.1 Algorithms of Time–Space Aggregation of Regeneration Times Based on Total or Partial Removing of Virtual Transitions 11.4.1.1 Reduced Recurrent Algorithms Time–Space Aggregation for Regeneration Times and Reduced Sets of Perturbation Conditions. It is useful to note that the perturbation conditions J1 , K1 , and R can be omitted from the conditions of Theorems 11.1 and 11.2 and, accordingly, the conditions k¯ n J1 , k¯ n K1 , k¯ n R, or    k¯ n J1 , k¯ n K1 , k¯ n R should be excluded from the statements (i) and (iii), respectively, of Theorems 11.1 or 11.2. Indeed, let the random variable ηε and the stochastic triplets ξ¯ε,i,n = ξε,i,n (t), t ≥ 0 , κε,i,n, ηε,i,n , i ∈ X = {1, . . . , m}, n = 1, 2, . . ., be defined for each ε ∈ (0, 1] and satisfy the following assumptions: (1) the random variable ηε takes values in space X and P{ηε = i} = pε,i, i ∈ X; (2) the process ξε,i,n (t) = ηε,i,n, t ≥ 0, for i ∈ X, n = 1, 2, . . .; (3) the random variables κε,i,n, i ∈ X, n = 1, 2, . . . take values in the interval [0, ∞) and P{κε,i,n ≤ t} = 1 − e−λε, i t , t ≥ 0, for i ∈ X, n = 1, 2, . . ., where λε,i = λ0,i u−1 ε,i, λ0,i > 0, uε,i ∈ (0, ∞); (4) the random variables ηε,i,n, i ∈ X, n = 1, 2, . . . take values in the space X and P{ηε,i,n = j} = pε,i j , j ∈ X, for i ∈ X, n = 1, 2, . . .; (5) the random variables ηε, κε,i,n, ηε,i,n, i ∈ X, n = 1, 2, . . . are mutually independent. It is obvious that random variables ηε and the triplets ξ¯ε,i,n = ξε,i,n (t), t ≥ 0 , κε,i,n, ηε,i,n , i ∈ X, n = 1, 2, . . . satisfy the model assumptions (K)–(O) formulated in Sect. 10.1.1.1. This makes it possible to define the Markov renewal process (ηε,n, κε,n ), using the relations (10.1) and (10.2), and the multi-alternating regenerative process (ξε (t), ηε (t)), using the relation (10.3). In this case, the Markov renewal process (ηε,n, κε,n ) has transition probabilities, Q ε,i j (t) = P{ηε,1 = j, κε,1 ≤ t/ηε,0 = i} = P{ηε,i,1 = j, κε,i,1 ≤ t} = pε,i j (1 − e−λε, i t ), t ≥ 0, i, j ∈ X.

(11.26)

Moreover, in this case the regenerative component ξε (t) and the modulating semi-Markov process ηε (t) coincide, i.e., for i ∈ X,

11.4 Comparison of algorithms of aggregation for regeneration times

Pi {ξε (t) = ηε (t), t ≥ 0} = 1.

271

(11.27)

Let us assume that the conditions G1 H1 , I1 , IH,1 , L1 , and LH,1 are satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). It is useful noting the condition G1 (b) can be omitted, since it holds due to the above assumption (3). In this case, the conditions J1 , K1 , R are automatically satisfied. The distribution function Fε,i j (t) = 1 − e−λε, i t , t ≥ 0, for i, j ∈ X. Thus, the condition J1 is satisfied, with the sets Y1,i = X, i ∈ X and the limiting distribution function F0,i j (t) = 1 − e−λ0, i t , t ≥ 0, for i, j ∈ X. −1 , for i, j ∈ X. Thus, the condition K is satisfied, Also, the expectation fε,i j = λε,i 1 −1 , for i, j ∈ X. with the limiting expectation f0,i j = λ0,i Finally, the functions qε,i (t, A) = I(i ∈ A)e−λε, i t , t ≥ 0, A ∈ BX, i ∈ X, for ε ∈ [0, 1]. Thus, the condition R is satisfied, with the corresponding limiting function q0,i (t, A) = I(i ∈ A)e−λ0, i t , t ≥ 0, A ∈ BX, i ∈ X and the corresponding set of locally uniform convergence U[q ·,i (·u ·,i, A)] = [0, ∞), for A ∈ BX, i ∈ X. The conditions k¯ n M and k¯ n M are formulated in terms of the transition probabilities pε,i j , i, j ∈ X, l = 1, . . . , n, and the normalisation functions u˜ε,i, i ∈ X, using the recurrent relations (11.8)–(11.11) and (11.17)–(11.21). Here, the conditions J1 , K1 , R are not used. 11.4.1.2 Recurrent Algorithms of Time–Space Aggregation for Regeneration Times Based on Total and Partial Removing of Virtual Transitions. Let us compare the conditions k¯ n M1 and k¯ n M and the normalisation functions used in the algorithms of time–space aggregation for regeneration times based on total and partial removal of virtual transitions from trajectories of the corresponding modulating semi-Markov processes.  ,  Let k¯ n−1 u˜ε,i, k¯ n uε,i k¯ n u˜ε,i k¯ n uε,i , n = 1, . . . , m − 1 be the functions given by the recurrent relations (11.10), (11.11), (11.20), and (11.21). The following lemma takes place. Lemma 11.1 Let the conditions G1 , H1 , I1 , IH,1 , L1 , and LH,1 be satisfied. Then: (i) The following relations connect the normalisation functions k¯ n−1 u˜ε,i , k¯ n uε,i ,   ¯k n u˜ ε,i , k¯ n uε,i, ε ∈ (0, 1] and n = 1, . . . , m − 1,  k¯ n u˜ ε,i  k¯ n u˜ ε,i

≤

k¯ n−1 u˜ ε,i,

for i ∈

k¯ n−1 X

\

k¯ n−1 K[k n ],

=

k¯ n−1 u˜ ε,i,

for i ∈

k¯ n−1 K[k n ],

(11.28)

and  k¯ n uε,i  k¯ n uε,i

where

≤

k¯ n uε,i,

for i ∈

k¯ n X

\

k¯ n−1 K[k n ],

=

k¯ n uε,i,

for i ∈

k¯ n X

∩

k¯ n−1 K[k n ],

(11.29)

272

11 Time-space aggregation of regeneration times k¯ n−1 K[k n ]

= {i ∈

k¯ n X

:

 k¯ n−1 p1,ik n

> 0} ∪ {k n }

= {i ∈

k¯ n X

:

k¯ n−1 p1,ik n

> 0} ∪ {k n }.

(11.30)

(ii) For any n = 1, . . . , m − 1, from the condition k¯ n M it follows that the condition  M is satisfied, and the following equalities hold for limits appearing in these k¯ n conditions:  k¯l−1 w0,kl ,i

= lim

 k¯l u˜ ε,kl ε→0 k¯ u˜  l ε,i

= lim

=

k¯l−1 w0,kl ,i,

for i ∈

k¯l−1 u˜ ε,kl

ε→0 k¯ u˜ ε,i l−1 k¯l−1 K[k l ], l

= 1, . . . , n.

(11.31)

Proof The relations (11.9), (11.11), (11.18), and (11.21) imply that, for i, j ∈ k¯0 pε,i j

=

 k¯0 pε,i j

k¯0 uε,i

=

 k¯0 uε,i

and

k¯0 X,

= pε,i j ,

(11.32)

= uε,i .

(11.33)

The relation (11.32) implies that, k¯0 K[k 1 ]

= {i ∈

k¯1 X

:

 k¯0 p1,ik1

> 0} ∪ {k1 }

= {i ∈

k¯1 X

:

k¯0 p1,ik1

> 0} ∪ {k1 }.

(11.34)

Thus, relation (11.30), given in the statement (i), takes place for n = 1. The conditions G1 , H1 and Theorems 11.1, 11.2 imply that, for i ∈ k¯l X, l = 0, . . . , m − 2,  (11.35) k¯l p¯ε,ii, k¯l p¯ε,ii > 0, ε ∈ (0, 1]. The relations (11.8) and (11.9) imply that, for i, j ∈ k¯0 p˜ε,i j

= I( j  i)

k¯0 pε,i j k¯0 p¯ε,ii

k¯0 X,

.

Also, the relations (11.8) and (11.9) imply that, for i ∈ k¯1 pε,i j

and, for i ∈

k¯1 X

∩

k¯0 K[k 1 ],

j∈

k¯1 pε,i j

(11.36) k¯1 X

j∈

k¯1 X,

k¯0 p˜ε,i j

(11.37)

+

k¯0 p˜ε,ik1 k¯0 p˜ε,k1 j .

(11.38)

k¯1 X,

=

k¯0 p˜ε,i j

k¯0 u˜ ε,i k¯1 X,

k¯0 K[k 1 ],

=

It was taken into account above that k¯0 p˜ε,ik1 = 0, for i ∈ The relations (11.10), (11.11) imply that, for i ∈ k¯0 X,

and, for i ∈

\

=

−1 k¯0 p¯ε,ii k¯0 uε,i,

k¯1 X

\

k¯0 K[k 1 ].

(11.39)

11.4 Comparison of algorithms of aggregation for regeneration times k¯1 uε,i

=

273

k¯0 u˜ ε,i .

(11.40)

The relations (11.17), (11.18), and (11.32) imply that, for i ∈ X, k¯0   k¯1 p˜ε,i j = k¯0 pε,i j = k¯0 pε,i j , and, for i ∈

k¯0 X

∩

k¯0 K[k 1 ],

 k¯1 p˜ε,i j

i∈

j∈

= I( j 

\

k¯0 K[k 1 ],

j∈

(11.41)

k¯0 X,  k¯0 pε,i j i)  k¯0 p¯ε,ii

= I( j  i)

k¯0 pε,i j k¯0 p¯ε,ii

=

k¯0 p˜ε,i j .

(11.42)

Also, the relations (11.20), (11.21), and (11.32), (11.41), (11.42) imply that, for k¯1 X \ k¯0 K[k 1 ], j ∈ k¯1 X,  k¯1 pε,i j

and, for i ∈

k¯1 X

∩

k¯0 K[k 1 ],  k¯1 pε,i j

i∈

k¯0 X

j∈

=

 k¯0 pε,i j

=

k¯0 pε,i j ,

(11.43)

k¯1 X,

=

 k¯1 p˜ε,i j

+

  k¯1 p˜ε,ik1 k¯1 p˜ε,k1 j

=

k¯0 p˜ε,i j

+

k¯0 p˜ε,ik1 k¯0 p˜ε,k1 j

=

k¯1 pε,i j .

(11.44)

Also, the relations (11.20), (11.21), and (11.32), (11.33), (11.39) imply that, for k¯0 X \ k¯0 K[k 1 ],   (11.45) k¯1 u˜ ε,i = k¯0 uε,i = k¯0 uε,i ≤ k¯0 u˜ ε,i,

and, for i ∈

k¯0 X

∩

k¯0 K[k 1 ],  k¯1 u˜ ε,i

=

−1  k¯0 p¯ε,ii k¯0 uε,i

=

−1 k¯0 p¯ε,ii k¯0 uε,i

=

k¯0 u˜ ε,i .

Also, the relations (11.20) and (11.21) imply that, for i ∈  k¯1 uε,i

=

(11.46)

k¯1 X,

 k¯1 u˜ ε,i .

(11.47)

The relations (11.34), (11.45)–(11.47) imply that the statement (i) holds for n = 1. The relation (11.46) and the condition k¯ 1 M imply that, for i ∈ k¯0 X ∩ k¯0 K[k1 ],  k¯0 w0,k1,i

 k¯1 u˜ ε,k1 ε→0 k¯ u˜  1 ε,i

= lim = lim

k¯0 u˜ ε,k1

ε→0 k¯ u˜ ε,i 0

=

k¯0 w0,k1,i

∈ [0, ∞).

(11.48)

The relation (11.48) implies that the condition k¯ 1 M holds, and, therefore, the statement (ii) holds for n = 1. The relations (11.8), (11.9), (11.17), (11.18), and (11.41), (11.42) imply that the following relations take place, for i ∈ k¯1 X \ k¯0 K[k1 ], j ∈ k¯1 X,

274

11 Time-space aggregation of regeneration times k¯1 pε,i j

=

k¯0 p˜ε,i j k¯0 pε,i j

= I( j  i) and, for i ∈

k¯1 X

∩

k¯0 K[k 1 ],

j∈

=

k¯1 pε,i j

 k¯0 pε,i j  , k¯0 p¯ε,ii

(11.49)

k¯1 X,

k¯0 p˜ε,i j  k¯0 p˜ε,i j

=

k¯0 p¯ε,ii

= I( j  i)

+ +

k¯0 p˜ε,ik1 k¯0 p˜ε,k1 j   k¯0 p˜ε,ik1 k¯0 p˜ε,k1 j

=

 k¯1 pε,i j .

(11.50)

The relations (11.32), (11.35)–(11.37), (11.41), (11.49), and (11.50) imply that {i ∈

k¯2 X

\

k¯0 K[k 1 ]

:

 k¯1 p1,ik2

> 0}  k¯0 p1,ik2

> 0}

k¯0 p1,ik2 k¯0 p1,ik2

> 0}

= {i ∈

k¯2 X

\

k¯0 K[k 1 ]

:

= {i ∈

k¯2 X

\

k¯0 K[k 1 ]

:

= {i ∈

k¯2 X

\

k¯0 K[k 1 ]

:

= {i ∈

k¯2 X

\

k¯0 K[k 1 ]

:

k¯0 p˜1,ik2

> 0}

= {i ∈

k¯2 X

\

k¯0 K[k 1 ]

:

k¯1 p1,ik2

> 0},

> 0}

k¯0 p¯1,ii

(11.51)

and {i ∈

k¯2 X

∩

= {i ∈

k¯0 K[k 1 ] k¯2 X

∩

:

 k¯1 p1,ik2

k¯0 K[k 1 ]

:

> 0} k¯1 p1,ik2

> 0}.

(11.52)

The relations (11.51) and (11.52) imply that the following relation takes place: k¯1 K[k 2 ]

 k¯1 p1,ik2

= {i ∈

k¯2 X

:

= {i ∈

k¯2 X

\ K[k1 ] :

∪ {i ∈ = {i ∈

k¯2 X

∪ {i ∈ = {i ∈

k¯2 X

\

k¯2 X

k¯2 X

:

∩

 k¯1 p1,ik2

k¯0 K[k 1 ]

k¯0 K[k 1 ]

∩

> 0} ∪ {k2 }

:

 k¯1 p1,ik2

k¯1 p1,ik2

k¯0 K[k 1 ]

k¯1 p1,ik2

:

> 0}

:

> 0} ∪ {k2 }

> 0}

k¯1 p1,ik2

> 0} ∪ {k2 },

> 0} ∪ {k2 } (11.53)

i.e., the relation (11.30), given in the statement (i), holds for n = 2. The relations (11.8), (11.9) and (11.34), (11.49), (11.53) imply that the following relations take place: for i ∈ (k¯1 X \ k¯0 K[k1 ], j ∈ k¯1 X,

11.4 Comparison of algorithms of aggregation for regeneration times k¯1 p˜ε,i j

and, for i ∈

k¯1 X

∩

k¯0 K[k 1 ],

k¯1 pε,i j

= I( j  i) = I( j  i)

j∈

275

k¯1 p¯ε,ii k¯0 pε,i j

1−

=

k¯0 pε,ii

k¯0 p˜ε,i j ,

(11.54)

k¯1 X,

k¯1 p˜ε,i j

= I( j  i)

k¯1 pε,i j k¯1 p¯ε,ii

.

(11.55)

The relations (11.8), (11.9) and (11.34), (11.53), (11.54) imply that the following relations take place, for i ∈ (k¯2 X \ k¯0 K[k1 ]) \ k¯1 K[k2 ], j ∈ k¯2 X, k¯2 pε,i j

and, for i ∈ (k¯2 X ∩

k¯0 K[k 1 ])

\

=

k¯1 p˜ε,i j

k¯1 K[k 2 ], k¯2 pε,i j

=

k¯0 p˜ε,i j ,

j∈

k¯2 X,

=

k¯1 p˜ε,i j .

(11.56)

(11.57)

Also, the relations (11.8), (11.9) and (11.34), (11.53), (11.54) imply that the following relations take place, for i ∈ (k¯2 X \ k¯0 K[k1 ]) ∩ k¯1 K[k2 ], j ∈ k¯2 X, k¯2 pε,i j

=

k¯1 p˜ε,i j

and, for i ∈ (k¯2 X ∩

+

=

k¯1 p˜ε,ik2 k¯1 p˜ε,k2 j

k¯0 K[k 1 ])

∩

k¯2 pε,i j

k¯1 K[k 2 ],

=

k¯1 p˜ε,i j

k¯0 p˜ε,i j

j∈ +

+

k¯0 p˜ε,ik2 k¯0 p˜ε,k2 j ,

(11.58)

k¯2 X,

k¯1 p˜ε,ik2 k¯1 p˜ε,k2 j .

(11.59)

The relations (11.10), (11.11) and (11.39), (11.40) imply that, for i ∈ k¯1 X \ K[k 1 ], k¯0 −1 −1 (11.60) k¯1 u˜ ε,i = k¯1 p¯ε,ii · k¯1 uε,i = k¯0 p¯ε,ii k¯0 uε,i = k¯0 u˜ ε,i, and, for i ∈

k¯1 X

∩

k¯0 K[k 1 ], k¯1 u˜ ε,i

=

−1 k¯1 p¯ε,ii

·

k¯1 uε,i

=

−1 −1 k¯1 p¯ε,ii k¯0 p¯ε,ii k¯0 uε,i .

(11.61)

It was taken into account above that k¯1 p¯ε,ii = 1, for i ∈ k¯1 X \ k¯0 K[k1 ]. Also, the relations (11.10) and (11.11) imply that, for i ∈ k¯2 X, k¯2 uε,i

=

k¯1 u˜ ε,i .

(11.62)

The relations (11.17), (11.18) and (11.32), (11.34), (11.41), (11.42), (11.53) imply that the following relations take place, for i ∈ (k¯1 X \ k¯0 K[k1 ]) \ k¯1 K[k2 ], j ∈ k¯1 X,  k¯2 p˜ε,i j

and, for i ∈ (k¯1 X ∩

k¯0 K[k 1 ])

\

=

 k¯1 pε,i j

k¯1 K[k 2 ],

=

 k¯0 pε,i j

j∈

k¯1 X,

=

k¯0 pε,i j ,

(11.63)

276

11 Time-space aggregation of regeneration times  k¯2 p˜ε,i j

=

 k¯1 p˜ε,i j

=

 k¯0 p˜ε,i j

=

k¯0 p˜ε,i j .

(11.64)

Also, the relations (11.10), (11.11), (11.17), (11.18) and (11.32), (11.34), (11.41), (11.42), (11.50), (11.53) imply that the following relations take place, for i ∈ (k¯1 X \ k¯0 K[k1 ]) ∩ k¯1 K[k2 ], j ∈ k¯1 X,  k¯2 p˜ε,i j

= I( j  i) = I( j  i)

and, for i ∈ (k¯1 X ∩

k¯0 K[k 1 ])

∩

 k¯2 p˜ε,i j

 k¯1 pε,i j  k¯1 p¯ε,ii k¯0 pε,i j k¯0 p¯ε,ii

k¯1 K[k 2 ],

j∈

= I( j  i) = I( j  i)

= I( j  i) =

 k¯0 pε,i j  k¯0 p¯ε,ii

k¯0 p˜ε,i j ,

(11.65)

k¯1 X,

 k¯1 pε,i j  k¯1 p¯ε,ii k¯1 pε,i j

=

k¯1 p¯ε,ii

k¯1 p˜ε,i j .

(11.66)

The relations (11.10), (11.11), (11.17), (11.18) and (11.32), (11.34), (11.41), (11.42), (11.50), (11.53), (11.63), (11.64) imply that the following relations take place, for i ∈ (k¯2 X \ k¯0 K[k1 ]) \ k¯1 K[k2 ], j ∈ k¯2 X,  k¯2 pε,i j

and for i ∈ (k¯2 X ∩

k¯0 K[k 1 ])

\

=

 k¯1 pε,i j

k¯1 K[k 2 ],

 k¯2 pε,i j

=

=

 k¯0 pε,i j

j∈

k¯2 X,

 k¯1 pε,i j

=

=

k¯0 pε,i j ,

(11.67)

k¯1 pε,i j .

(11.68)

Also, the relations (11.10), (11.11), (11.17), (11.18) and (11.32), (11.34), (11.41), (11.42), (11.50), (11.53), (11.63), (11.64), (11.65), (11.66) imply that the following relations take place, for i ∈ (k¯2 X \ k¯0 K[k1 ]) ∩ k¯1 K[k2 ], j ∈ k¯2 X,  k¯2 pε,i j

and, for i ∈ (k¯2 X ∩

k¯0 K[k 1 ])  k¯2 pε,i j

=

∩

=

 k¯2 p˜ε,i j

+

  k¯2 p˜ε,ik2 k¯2 p˜ε,k2 j

=

k¯0 p˜ε,i j

+

k¯0 p˜ε,ik2 k¯0 p˜ε,k2 j ,

k¯1 K[k 2 ],

 k¯1 p˜ε,i j

+

j∈

(11.69)

k¯2 X,

  k¯1 p˜ε,ik2 k¯1 p˜ε,k2 j

=

k¯2 pε,i j .

(11.70)

The relations (11.20), (11.21), and (11.32), (11.33), (11.39), (11.45), (11.46), (11.60), (11.61) imply that, for i ∈ (k¯1 X \ k¯0 K[k1 ]) \ k¯1 K[k2 ],  k¯2 u˜ ε,i

and, for i ∈ (k¯1 X ∩

=

 k¯1 uε,i

k¯0 K[k 1 ])

\

=

 k¯1 u˜ ε,i

k¯1 K[k 2 ],

=

 k¯0 uε,i

≤

k¯0 u˜ ε,i

=

k¯1 u˜ ε,i ,

(11.71)

11.4 Comparison of algorithms of aggregation for regeneration times  k¯2 u˜ ε,i

=

 k¯1 uε,i

 k¯1 u˜ ε,i

=

=

−1 k¯0 p¯ε,ii k¯0 uε,i

≤

277 k¯1 u˜ ε,i .

(11.72)

Also, the relations (11.20), (11.21), and (11.32), (11.33), (11.39), (11.45), (11.46), (11.60), (11.61) imply that, for i ∈ (k¯1 X \ k¯0 K[k1 ]) ∩ k¯1 K[k2 ],  k¯2 u˜ ε,i

= =

and, for i ∈ (k¯1 X ∩

−1  k¯1 p¯ε,ii k¯1 uε,i −1 k¯0 p¯ε,ii k¯0 uε,i

=

−1  k¯0 p¯ε,ii k¯0 uε,i

=

k¯1 u˜ ε,i ,

k¯0 K[k 1 ])

∩

k¯1 K[k 2 ],

 k¯2 u˜ ε,i

=

−1  k¯1 p¯ε,ii k¯1 uε,i −1 −1 k¯1 p¯ε,ii k¯0 p¯ε,ii k¯0 uε,i

=

=

(11.73)

k¯1 u˜ ε,i .

Also, the relations (11.20), (11.21) imply that, for i ∈  k¯2 uε,i

=

(11.74)

k¯2 X,

 k¯2 u˜ ε,i .

(11.75)

The relations (11.34), (11.53), (11.71)–(11.74) imply that the statement (i) holds for n = 2. The relations (11.73), (11.74) and the condition k¯ 2 M imply that, for i ∈ k¯1 X ∩ ¯k1 K[k 2 ],  k¯1 w0,k2,i

 k¯2 u˜ ε,k2 ε→0 k¯ u˜  2 ε,i

= lim = lim

k¯1 u˜ ε,k2

ε→0 k¯ u˜ ε,i 1

=

k¯1 w0,k2,i

∈ [0, ∞).

(11.76)

The relations (11.48) and (11.76) imply that condition k¯ 2 M holds, and, therefore, the statement (ii) holds for n = 2. The proof of the statements (i) and (ii) for n = 3, . . . , m − 1 is similar. 

11.4.2 Asymptotic Communicative Structure of Phase Spaces for Modulating Semi-Markov Processes for Embedded Alternating Regenerative Processes 11.4.2.1 Embedded Alternating Regenerative Processes Based on Total Removing of Virtual Transitions. We obtain ergodic theorems for perturbed multialternating regenerative processes (ξε (t), ηε (t)) by reducing them to ergodic theorems for perturbed embedded alternating regenerative processes (k¯ m−2 ξε (t), k¯ m−2 ηε (t)) or to ergodic theorems for perturbed embedded regenerative processes (k¯ m−1 ξε (t), k¯ m−1 ηε (t)).

278

11 Time-space aggregation of regeneration times

We also use an alternative method and obtain ergodic theorems for the perturbed multi-alternating regenerative processes (ξε (t), ηε (t)) by reducing them to ergodic theorems for perturbed embedded alternating regenerative processes (k¯ m−2 ξε (t), k¯ m−2 ηε (t)) or to ergodic theorems for perturbed embedded regenerative processes (k¯ m−1 ξε (t), k¯ m−1 ηε (t)). In both cases, we apply to these embedded processes ergodic theorems presented in Part 1. The corresponding ergodic theorems for perturbed multi-alternating regenerative processes are presented in Chap. 13. Let us consider the first alternative, where the embedded alternating regenerative processes (k¯ m−2 ξε (t), k¯ m−2 ηε (t)) and the embedded regenerative processes (k¯ m−1 ξε (t), k¯ m−1 ηε (t)) are used. In this case, the corresponding modulating semi-Markov process k¯ m−2 ηε (t) has the two-state phase space k¯ m−2 X = {k m−1, k m }, while the modulating semi-Markov process k¯ m−1 ηε (t) has the one-state phase space k¯ m−1 X = {k m }. We assume that the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 are satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)) and use the reduced form of Theorem 11.1 described in Sect. 11.4.1.1. In the case of using the embedded alternating regenerative processes (k¯ m−2 ξε (t), k¯ m−2 ηε (t)), it is also assumed that the condition k¯ m−2 M is satisfied. In this case, the states k m−1 and k m are the two most absorbing states in the initial phase space k¯0 X = X. However, the states k m−1 and k m themselves are not ordered by their absorbing rates. In the case of using embedded regenerative processes (k¯ m−1 ξε (t), k¯ m−1 ηε (t)), it is also assumed that the condition k¯ m−1 M is satisfied. In this case, state k m is the most absorbing state in the initial phase space k¯0 X = X. Let us find out the asymptotic communicative structure of the phase space k¯ m−2 X for the embedded modulating semi-Markov processes k¯ m−2 ηε (t). Consider the semi-Markov process k¯ m−3 η˜ε (t), which is the result of the total removal of virtual transitions from trajectories of the reduced semi-Markov process k¯ m−3 ηε (t). This process has the phase space k¯ m−3 X = {k m−2, k m−1, k m }. The semi-Markov process k¯ m−2 ηε (t) is constructed by exclusion of the state k m−2 from the phase space k¯ m−3 X. According to Theorem 11.1 and the remarks made ˜ , k¯ H ˜ , k¯ I˜ , k¯ I˜ , k¯ L ˜ , k¯ L ˜ in Sect. 11.2, the conditions k¯ G m−3

1

m−3

1

m−3

1

m−3

H,1

m−3

1

m−3

H,1

are satisfied for the semi-Markov processes k¯ m−3 η˜ε (t) and the conditions k¯ m−2 G1 , k¯ m−2 H1 , k¯ m−2 I1 , k¯ m−2 IH,1 , k¯ m−2 L1 , k¯ m−2 LH,1 are satisfied for the semi-Markov processes k¯ m−2 ηε (t). Note that, according to Theorem 11.1, the conditions k¯ m−3 G1 , k¯ m−3 H1 , k¯ m−3 I1 are satisfied. Thus, the phase space k¯ m−3 X is, for every ε ∈ (0, 1], one class of communicative states for the embedded Markov chain k¯ m−3 η˜ε,n . Moreover, either the transition probabilities k¯ m−3 p˜ε,i j > 0, ε ∈ (0, 1] or k¯ m−3 p˜ε,i j = 0, ε ∈ (0, 1], for i, j ∈k¯ m−3 X and k¯ m−3 p˜ε,i j → k¯ m−3 p˜ε,i j as ε → 0, for i, j ∈k¯ m−3 X. Also, according to Theorem 11.1, the conditions k¯ m−2 G1 , k¯ m−2 H1 , k¯ m−2 I1 are satisfied. Thus, the phase space k¯ m−2 X is, for every ε ∈ (0, 1], one class of communicative states for the embedded Markov chain k¯ m−2 ηε,n . Moreover, either the transition prob-

11.4 Comparison of algorithms of aggregation for regeneration times

279

abilities k¯ m−2 pε,i j > 0, ε ∈ (0, 1] or k¯ m−2 pε,i j = 0, ε ∈ (0, 1], for i, j ∈k¯ m−2 X and k¯ m−2 pε,i j → k¯ m−2 pε,i j as ε → 0, for i, j ∈k¯ m−2 X. Let k¯ m−3 η˜0,n be a Markov chain with the phase space k¯ m−3 X and transition probabilities k¯ m−3 p˜0,i j , i, j ∈ k¯ m−2 X, and k¯ m−2 η0,n be a Markov chain with the phase space k¯ m−2 X and transition probabilities k¯ m−2 p0,i j , i, j ∈ k¯ m−2 X. The transition probabilities for the embedded Markov chains k¯ m−3 η˜ε,n and k¯ m−2 ηε,n are connected by the following relations: k¯ m−2 pε,i j

=

k¯ m−3 p˜ε,i j

+

k¯ m−3 p˜ε,ik m−2 k¯ m−3 p˜ε,k m−2 j ,

for i, j ∈

k¯ m−2 X.

(11.77)

Note that the Markov chain k¯ m−3 η˜ε,n does not make virtual transitions, for every ε ∈ (0, 1], i.e., (11.78) k¯ m−3 p˜ε,ii = 0, for i ∈ k¯ m−3 X. The relation (11.77) implies that the Markov chain virtual transitions, i.e., k¯ m−3 p˜0,ii

= 0, for i ∈

k¯ m−3 η0,n

k¯ m−3 X.

also does not make (11.79)

Three cases for the values of the limiting transitions probabilities k¯ m−3 p0,km−1 km and k¯ m−3 p˜0,km km−1 should be analysed. (1) k¯ m−3 p˜0,km−1 km , k¯ m−3 p˜0,km km−1 > 0. Then, both limiting transitions probabilities k¯ m−2 p0,km−1 km , k¯ m−2 p0,km km−1 > 0. Thus, the phase space k¯ m−2 X is one class of communicative state for the Markov chain k¯ m−2 η0,n . This case corresponds to the model with regular perturbations for the embedded modulating semi-Markov processes k¯ m−2 ηε (t). (2) k¯ m−3 p˜0,km−1 km > 0, k¯ m−3 p˜0,km km−1 = 0 or k¯ m−3 p˜0,km−1 km = 0, k¯ m−3 p˜0,km km−1 > 0. For example, let the first alternative takes place. Then, the limiting transitions probability k¯ m−2 p0,km−1 km > 0, while the limiting transition probability k¯ m−2 p0,km km−1 > 0, if k¯ m−3 p˜ε,km−2 km−1 > 0, or k¯ m−2 p0,km km−1 = 0, if k¯ m−3 p˜ε,km−2 km−1 = 0. It follows from the relation (11.77), since, according to the relation (11.79), k¯ m−3 p˜ε,km km−2 = 1 − k¯ m−3 p˜0,km km−1 = 1. In the first case, the phase space k¯ m−2 X is one class of communicative states for the Markov chain k¯ m−2 η0,n . In the second case, the phase space k¯ m−2 X is the union of the closed class of communicative states {k m } and the class of transient states {k m−1 }. The first case corresponds to the model with regular perturbations, while the second case corresponds to the model with semi-regular perturbations for the embedded modulating semi-Markov processes k¯ m−2 ηε (t). (3) k¯ m−3 p˜0,km−1 km , k¯ m−3 p˜0,km km−1 = 0. Then, both limiting transitions probabilities k¯ m−2 p0,km−1 km , k¯ m−2 p0,km km−1 > 0, if ¯k m−3 p˜ε,k m−2 k m−1 ∈ (0, 1), or k¯ m−2 p0,k m−1 k m > 0, k¯ m−2 p0,k m k m−1 = 0, if k¯ m−3 p˜ε,k m−2 k m = 1, or k¯ m−2 p0,km−1 km = 0, k¯ m−2 p0,km km−1 > 0, if k¯ m−3 p˜ε,km−2 km = 0. It follows from the relation (11.77), since, according to the relation (11.79), k¯ m−3 p˜ε,km km−2 , k¯ m−3 p˜0,k m−1 k m−2 = 1. In the first case, the phase space k¯ m−2 X is one class of communicative states for the Markov chain k¯ m−2 η0,n . In the second case, the phase space k¯ m−2 X is the union

280

11 Time-space aggregation of regeneration times

of the closed class of communicative states {k m } and the class of transient states {k m−1 }, and in the third case, the phase space k¯ m−2 X is the union of the closed class of communicative states {k m−1 } and the class of transient states {k m }. The first case corresponds to the model with regular perturbations, while the second and third cases correspond to the model with semi-regular perturbations for the embedded modulating semi-Markov processes k¯ m−2 ηε (t). It follows from the above analysis that the case with both limiting transitions probabilities k¯ m−2 p0,km−1 km , k¯ m−2 p0,km km−1 = 0, which would correspond to the model with singular perturbations for the embedded modulating semi-Markov processes k¯ m−2 ηε (t), is not possible. The following lemma summarises the above remarks. Lemma 11.2 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , and k¯ m−2 M be satisfied. Then, the conditions k¯ m−2 G1 , k¯ m−2 H1 , k¯ m−2 I1 , k¯ m−2 IH,1 , k¯ m−2 L1 , k¯ m−2 LH,1 are satisfied, and the embedded Markov chains k¯ m−2 ηε,n are regularly perturbed (k¯ m−2 p0,km−1 km , k¯ m−2 p0,k m k m−1 > 0), or semi-regularly perturbed (k¯ m−2 p0,k m−1 k m > 0, k¯ m−2 p0,k m k m−1 = 0 or k¯ m−2 p0,km−1 km = 0, k¯ m−2 p0,km km−1 > 0). Lemma 11.2 imposes some restrictions on ergodic theorems based on the use of embedded alternating regenerative processes (k¯ m−2 ξε (t), k¯ m−2 ηε (t)). Only ergodic theorems for regularly and semi-regularly perturbed alternating regenerative processes presented in Chaps. 4 and 5 can be used to obtain ergodic theorems for multialternating regenerative processes. The ergodic theorems for singularly perturbed alternating regenerative processes presented in Chaps. 6–8 are not applicable. Alternatively, the embedded regenerative processes (k¯ m−1 ξε (t), k¯ m−1 ηε (t)) can be used. As mentioned above, in this case the modulating semi-Markov process k¯ m−1 ηε (t) has the one-state phase space k¯ m−1 X = {k m }. In this case, the ergodic theorems for perturbed regenerative processes presented in Chap. 2 can be applied. As we will show in Chap. 13, both approaches give almost the equivalent, socalled super-long time ergodic theorems for perturbed multi-alternating regenerative processes. They differ only in some features of the calculation of the corresponding limiting distributions. 11.4.2.2 Embedded Alternating Regenerative Processes Based on Partial Removing of Virtual Transitions. In this case, we obtain ergodic theorems for perturbed multi-alternating regenerative processes (ξε (t), ηε (t)) reducing them to ergodic theorems for perturbed embedded alternating regenerative processes (k¯ m−2 ξε (t), k¯ m−2 ηε (t)) or perturbed embedded regenerative processes (k¯ m−1 ξε (t),  k¯ m−1 ηε (t)). In this case, the corresponding modulating semi-Markov process k¯ m−2 ηε (t) has the two-state phase space k¯ m−2 X = {k m−1, k m }, while the modulating semi-Markov process k¯ m−1 ηε (t) has the one-state phase space k¯ m−1 X = {k m }. As above, we assume that conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 are satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)) and use the reduced form of Theorem 11.2 described in Sect. 11.4.1.1.

11.4 Comparison of algorithms of aggregation for regeneration times

281

In the case of using the embedded alternating regenerative processes (k¯ m−2 ξε (t),  k¯ m−2 ηε (t)), it is also assumed that the condition k¯ m−2 M (stronger than the condition  k¯ m−2 M ) is satisfied. In this case, the states k m−1 and k m are the two most absorbing states in the initial phase space k¯0 X = X. However, the states k m−1 and k m are not ordered by their absorbing rates. In the case of using the embedded regenerative processes (k¯ m−1 ξε (t), k¯ m−1 ηε (t)), it is also assumed that the condition k¯ m−1 M (stronger than the condition k¯ m−1 M) is satisfied. In this case, state k m is the most absorbing state in the initial phase space k¯0 X = X. Let us find out the asymptotic communicative structure of the phase space k¯ m−2 X for the embedded modulating semi-Markov processes k¯ m−2 ηε (t). Consider the semi-Markov process k¯ m−2 η˜ε (t), which is the result of the partial removal of virtual transitions from trajectories of the reduced semi-Markov process k¯ m−3 ηε (t). This process has the phase space k¯ m−3 X = {k m−2, k m−1, k m }. The semi-Markov process k¯ m−2 ηε (t) is constructed by the excluding the state k m−2 from the phase space k¯ m−3 X. According to Theorem 11.2 and Lemma 11.1, the   ˜  , k¯ H ˜ , ˜ , ˜  are satisfied for the conditions k¯ m−2 G I˜ , I˜ , L L 1 m−2 1 k¯ m−2 1 k¯ m−2 H,1 k¯ m−2 1 k¯ m−2 H,1  semi-Markov processes k¯ m−2 η˜ε (t) and the conditions k¯ m−2 G, k¯ m−2 H, k¯ m−2 I, k¯ m−2 IH ,    k¯ m−2 L , k¯ m−2 LH are satisfied for the semi-Markov processes k¯ m−2 ηε (t). ˜  , k¯ H ˜  , k¯ I˜  Note that, according to Theorem 11.2, the conditions k¯ m−2 G m−2 m−2 1 1 1 are satisfied. Thus, the phase space k¯ m−3 X is, for every ε ∈ (0, 1], one class of  . Moreover, either communicative states for the embedded Markov chain k¯ m−2 η˜ε,n   the transition probabilities k¯ m−2 p˜ε,i j > 0, ε ∈ (0, 1] or k¯ m−2 p˜ε,i j = 0, ε ∈ (0, 1], for   i, j ∈ k¯ m−3 X and k¯ m−2 p˜ε,i j → k¯ m−2 p˜ε,i j as ε → 0, for i, j ∈k¯ m−3 X. Also, according to Theorem 11.2, the conditions k¯ m−2 G1 , k¯ m−2 H1 , k¯ m−2 I1 are satisfied. Thus, the phase space k¯ m−2 X is, for every ε ∈ (0, 1], one class of communicative  . Moreover, either the transition probstates for the embedded Markov chain k¯ m−2 ηε,n   abilities k¯ m−2 pε,i j > 0, ε ∈ (0, 1] or k¯ m−2 pε,i j = 0, ε ∈ (0, 1], for i, j ∈k¯ m−2 X and   k¯ m−2 pε,i j → k¯ m−2 pε,i j as ε → 0, for i, j ∈k¯ m−2 X.  Let k¯ m−2 η˜0,n be a Markov chain with the phase space k¯ m−3 X and transition proba , i, j ∈ bilities k¯ m−2 p˜0,i k¯ m−2 X, and k¯ m−2 η0,n be a Markov chain with the phase space j  k¯ m−2 X and transition probabilities k¯ m−2 p0,i j , i, j ∈ k¯ m−2 X.  and  The transition probabilities for the embedded Markov chains k¯ m−2 η˜ε,n k¯ m−2 ηε,n are connected by the following formulas:

 k¯ m−2 pε,i j

 ⎧ ⎪ k¯ m−2 p˜ε,i j ⎪ ⎪ ⎨ ⎪ for i ∈ k¯ m−2 X \ k¯ m−3 K[k m−2 ], j ∈ k¯ m−2 X, =    ⎪ k¯ m−2 p˜ε,i j + k¯ m−2 p˜ε,ik m−2 k¯ m−2 p˜ε,k m−2 j ⎪ ⎪ ⎪ for i ∈ k¯ m−2 X ∩ k¯ m−3 K[k m−2 ], j ∈ k¯ m−2 X. ⎩

(11.80)

Two cases should be analysed, where: (4) set k¯ m−3 K[k m−2 ] = {k m−2, k m−1, k m } and (5) set k¯ m−3 K[k m−2 ] = {k m−2, k m−1 } or k¯ m−3 K[k m−2 ] = {k m−2, k m }. (4) k¯ m−3 K[k m−2 ] = {k m−2, k m−1, k m }.

282

11 Time-space aggregation of regeneration times

 does not make virtual transitions, for every In this case, the Markov chain k¯ m−2 η˜ε,n ε ∈ (0, 1], i.e.,  (11.81) k¯ m−2 p˜ε,ii = 0, for i ∈ k¯ m−3 X.

The relation (11.81) implies that the Markov chain virtual transitions, i.e.,  k¯ m−3 p˜0,ii

= 0, for i ∈

k¯ m−3 η0,n

also does not make

k¯ m−3 X.

(11.82)

The subsequent analysis is similar to that done in Sect. 11.4.2.1. Three cases are possible, where the phase space k¯ m−2 X of the Markov chain k¯ m−2 η0,n is one class of communicative states or is the union of the closed class of communicative states {k m } and the class of transient states {k m−1 } or is the union of the closed class of communicative states {k m−1 } and the class of transient states {k m }. The first case corresponds to the model with regular perturbations, while the second and third cases correspond to the model with semi-regular perturbations for the embedded modulating semi-Markov processes k¯ m−2 ηε (t).  , ¯ p The case with both limiting transitions probabilities k¯ m−2 p0,k m−1 k m k m−2 0,k m k m−1 = 0, which would correspond to the model with singular perturbations for embedded modulating semi-Markov processes k¯ m−2 ηε (t), is not possible. (5) k¯ m−3 K[k m−2 ] = {k m−2, k m−1 } or k¯ m−3 K[k m−2 ] = {k m−2, k m }. For example, let the first alternative takes place.  does not make In this case, for every ε ∈ (0, 1], the Markov chain k¯ m−2 η˜ε,n virtual transitions of the form i → i, for i ∈ k¯ m−3 K[k m−2 ], the transition of the form k m → k m is possible, while the transition k m → k m−2 is not possible. Thus,  0 for i = k m−2, k m−1,  (11.83) ¯k m−2 p˜ε,ii =  ∈ [0, 1] for i = k m . 1 − k¯ m−2 p˜ε,k ,k m m−1 The relation (11.83) implies that  0 for i = k m−2, k m−1,   k¯ m−2 p˜0,ii = 1 − ¯ p ˜ ∈ [0, 1] for i = k m . k m−2 0,k m,k m−1

(11.84)

As above, three cases are possible, where the phase space k¯ m−2 X of the Markov chain k¯ m−2 η0,n is one class of communicative states or is the union of the closed class of communicative states {k m } and the class of transient states {k m−1 } or is the union of the closed class of communicative states {k m−1 } and the class of transient states {k m }. The first case corresponds to the model with regular perturbations, while the second and third cases correspond to the model with semi-regular perturbations for the embedded modulating semi-Markov processes k¯ m−2 ηε (t). However, the fourth case is also possible, where the phase space k¯ m−2 X of the Markov chain k¯ m−2 η0,n is the union of two closed classes of communicative states {k m−1 } and {k m }.

11.4 Comparison of algorithms of aggregation for regeneration times

283

 It is the case where the limiting probabilities k¯ m−3 p˜0,k , ¯ p˜ =0 m,k m−1 k m−3 0,k m−1,k m  and k¯ m−3 p˜0,km−2,km−1 = 1 and, thus, according to the relations (11.80) and (11.84),  , ¯ p = 0. both limiting transition probabilities k¯ m−2 p0,k m,k m−1 k m−2 0,k m−1,k m This case corresponds to the model with singular perturbations for the embedded modulating semi-Markov processes k¯ m−2 ηε (t). The following lemma summarises the above remarks.

Lemma 11.3 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 and k¯ m−2 M be satisfied.   , k¯ m−2 L1 , k¯ m−2 LH,1 are satisfied, Then, the conditions k¯ m−2 G1 , k¯ m−2 H1 , k¯ m−2 I1 , k¯ m−2 IH,1   are regularly perturbed (k¯ m−2 p0,k , and the embedded Markov chains k¯ m−2 ηε,n m−1 k m    p > 0), or semi-regularly perturbed ( p > 0, p = k¯ m−2 0,k m k m−1 k¯ m−2 0,k m−1 k m k¯ m−2 0,k m k m−1    = 0, > 0), or singularly perturbed (k¯ m−2 p0,k , 0 or k¯ m−2 p0,k ¯ m−2 p0,k k k k m m−1 m−1 m m−1 k m  p = 0). k¯ m−2 0,k m k m−1  Remark 11.1 If the conditions of Lemma 11.3 hold, then the Markov chains k¯ m−2 ηε,n   can be singularly perturbed if and only if k¯ m−3 p˜0,km,km−1 , k¯ m−3 p˜0,km−1,km = 0 and:  = 1, or (b) k¯ m−3 K[k m−2 ] = (a) k¯ m−3 K[k m−2 ] = {k m−2, k m−1 } and k¯ m−3 p˜0,k m−2,k m−1  {k m−2, k m } and k¯ m−3 p˜0,km−2,km = 1.

Remark 11.2 Typical examples in which the assumptions formulated in Remark 11.1 can be fulfilled are presented by a model in which the initial semi-Markov processes ηε (t) are birth–death-type semi-Markov processes. In this case, the reduced semiMarkov processes k¯ n ηε (t), n = 1, . . . , m − 1, are also birth–death-type semi-Markov processes. In particular, the set k¯ m−3 K[k m−2 ] = {k m−2, k m−1 } or k¯ m−3 K[k m−2 ] = {k m−2, k m }, if the state k m−2 is one of the two end states in the phase space k¯ m−3 X = {k m−1, k m−1, k m } of the birth–death-type semi-Markov processes k¯ m−3 ηε (t). Such example is considered in Chap. 1. According to Lemma 11.3, the ergodic theorems for regularly, semi-regularly, and singularly perturbed alternating regenerative processes presented in Chaps. 4–8 can be applied in the case of using the embedded alternating regenerative processes (k¯ m−2 ξε (t), k¯ m−2 ηε (t)). Alternatively, the embedded regenerative processes (k¯ m−1 ξε (t), k¯ m−1 ηε (t)). As mentioned above, in this case the modulating semi-Markov process k¯ m−1 ηε (t) has the one-state phase space k¯ m−1 X = {k m }. In this case, the ergodic theorems for perturbed regenerative processes presented in Chap. 2 can be applied. As we show in Chap. 13, both approaches give almost equivalent super-long time ergodic theorems for perturbed multi-alternating regenerative processes. They differ only by some features of the calculation of the corresponding limiting distributions. Moreover, these super-long time ergodic theorems are also almost equivalent to super-long time ergodic theorems for perturbed multi-alternating regenerative processes, which can be obtained with the use of regularly and semi-regularly perturbed embedded alternating regenerative processes (k¯ m−2 ξε (t), k¯ m−2 ηε (t)) and perturbed embedded regenerative processes (k¯ m−1 ξε (t), k¯ m−1 ηε (t)).

284

11 Time-space aggregation of regeneration times

However, in the cases of singularly perturbed embedded alternating regenerative processes (k¯ m−2 ξε (t), k¯ m−2 ηε (t)), we can also obtain the so-called long and short time ergodic theorems for perturbed multi-alternating regenerative processes, which cannot be obtained with the use of regularly and semi-regularly perturbed embedded alternating regenerative processes (k¯ m−2 ξε (t), k¯ m−2 ηε (t)) and perturbed embedded regenerative processes (k¯ m−1 ξε (t), k¯ m−1 ηε (t)). Such theorems are presented in Sect. 13.2.

Chapter 12

Embedded Processes for Perturbed Multi-Alternating Regenerative Processes

In this chapter, we describe embedded alternating regenerative processes and embedded regenerative processes for perturbed multi-alternating regenerative processes. These embedded processes are constructed with the use of recurrent algorithms of time–space aggregation of regeneration times based on total and partial removing of virtual transition and reduction of phase space for modulating semi-Markov processes. This chapter includes two sections. In Sect. 12.1, we describe embedded alternating regenerative processes constructed with the use of recurrent algorithms of time–space aggregation of regeneration times based on total and partial removing of virtual transition and reduction of phase space for modulating semi-Markov processes. The corresponding perturbation conditions and recurrent formulas for computing characteristics of embedded alternating regenerative processes are given in Lemmas 12.1–12.8. In Sect. 12.2, we describe embedded regenerative processes constructed with the use of recurrent algorithms of time–space aggregation of regeneration times based on total and partial removing of virtual transition and reduction of phase space for modulating semi-Markov processes. The corresponding perturbation conditions and recurrent formulas for computing characteristics of embedded regenerative processes are given in Lemmas 12.9–12.16.

12.1 Embedded Alternating Regenerative Processes In this section, we describe embedded alternating regenerative processes built using recurrent algorithms of time–space aggregation of regeneration times based on total and partial removing of virtual transition and reduction of phase space for modulating semi-Markov processes.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0_12

285

286

12 Ergodic theorems for perturbed MARP

12.1.1 Embedded Alternating Regenerative Processes Based on Total Removing of Virtual Transitions 12.1.1.1 Embedded Alternating Regenerative Processes with Transition Period Based and Totally Removed Virtual Transitions. Let (ξε (t), ηε (t)) be, for every ε ∈ (0, 1], the multi-alternating regenerative process with the phase space Z × X defined in Sect. 10.1.1. We assume that the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , and R are satisfied for these processes. Also, let (ηε,n, κε,n ) be the corresponding Markov renewal process used to construct the modulating semi-Markov process ηε (t). According to the remarks made in Sect. 11.2.1.3, there exists a sequence of states k¯ n = k1, . . . , k m−2 in the phase space X = {1, . . . , m} of the modulating semiMarkov processes ηε (t) such that the condition k¯ m−2 M2 is satisfied. Let also (k¯ m−2 ξε (t), k¯ m−2 ηε (t)) be the multi-alternating regenerative process with the reduced phase space of modulating semi-Markov process k¯ m−2 X = X \ {k1, . . . , k m−2 } = {k m−1, k m } built using the recurrent algorithm of time–space aggregation of regeneration times (described in Sect. 11.2) applied to the multialternating regenerative process (ξε (t), ηε (t)). In this case, (k¯ m−2 ξε (t), k¯ m−2 ηε (t)) is an alternating regenerative process with a two-state modulating semi-Markov process k¯ m−2 ηε (t). The states k m−1, k m are the two most absorbing states in the phase space X of the original modulating semi-Markov processes ηε (t). It is useful to note that these two states are not compared in absorption rates. Let us assume that the initial distribution of the random variable ηε = ηε (0) satisfies the following condition: k¯ m−2 W: P {ηε (0)

∈

The condition

k¯ m−2 X}

k¯ m−2 W

= 1, for ε ∈ (0, 1].

implies that, for ε ∈ (0, 1], P{k¯ m−2 ηε (t) ∈ k¯ m−2 X, t ≥ 0} = 1.

(12.1)

Therefore, (k¯ m−2 ξε (t), k¯ m−2 ηε (t)) = (ξε (t), k¯ m−2 ηε (t)) is an alternating regenerative process, with a phase space Z × k¯ m−2 X. Let us now consider the general case when the condition k¯ m−2 W is not satisfied. We introduce the following stopping moments, for ε ∈ (0, 1],  0 if ηε,0 ∈ k¯ m−2 X, (12.2) k¯ m−2 αε = min(n ≥ 1 : η ∈ X) if ηε,0  k¯ m−2 X, ¯ ε,n k m−2 and k¯ m−2 τε =

k¯ m−2 αε



κε,n .

(12.3)

n=1

The process (ξε (t), ηε (t)) can be considered as a multi-alternating regenerative process with a transition period [0, k¯ m−2 τε ).

12.1 Embedded ARP

287

Let us consider the shifted multi-alternating regenerative process, (ξε(1) (t), ηε(1) (t)) = (ξε (k¯ m−2 τε + t), ηε (k¯ m−2 τε + t)), t ≥ 0.

(12.4)

Obviously, the condition k¯ m−2 W is satisfied for these multi-alternating regenerative processes. Let, also, (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)) be the multi-alternating regenerative process (with the reduced phase space k¯ m−2 X = {k m−1, k m } for the modulating semi-Markov

processes k¯ m−2 ηε(1) (t)) built using the recurrent algorithm for time–space aggregation of regeneration times (described in Sect. 11.2) applied to the multi-alternating regenerative process (ξε(1) (t), ηε(1) (t)). According to Theorem 11.1, the following relation takes place, for i ∈ k¯ m−2 X, t ≥ 0, A ∈ BZ and ε ∈ (0, 1], (1) (t, A) = Pi {ξε(1) (t) ∈ A} = Pi {k¯ m−2 ξε(1) (t) ∈ A}. Pε,i

(12.5)

(1) For each A ∈ BZ , the probabilities Pε,i (t, A), i ∈ k¯ m−2 X are measurable functions of t ≥ 0 and the only bounded solution of the following system of renewal type equations, (1) Pε,i (t, A) =

k¯ m−2 qε,i (t,



+

j ∈ k¯

m−2

A) ∫ t 0

X

(1) Pε, j (t − s, A) k¯ m−2 Q ε,i j (ds), t ≥ 0, i ∈

where, for A ∈ BZ, t ≥ 0, i, j ∈ k¯ m−2 qε,i (t,

k¯ m−2 X,

(12.6)

k¯ m−2 X,

A) = Pi {ξε (t) ∈ A,

k¯ m−2 ζε,1

> t},

(12.7)

and k¯ m−2 Q ε,i j (t)

= Pi { k¯ m−2 ζε,1 ≤ t,

k¯ m−2 ηε (k¯ m−2 ζε,1 )

= j}.

(12.8)

Also, the probabilities Pε,i (t, A) = Pi {ξε (t) ∈ A}, i ∈ X are, for every A ∈ BZ , the measurable functions of t ≥ 0, which satisfy the following renewal type relations, Pε,i (t, A) =

k¯ m−2 qε,i (t,

+



j ∈ k¯

m−2

X

A) ∫ t 0

(1)  Pε, j (t − s, A) k¯ m−2 Q ε,i j (ds), t ≥ 0, i ∈ X,

where, for A ∈ BZ, t ≥ 0, i ∈ X, j ∈ k¯ m−2 qε,i (t,

and

 k¯ m−2 Q ε,i j (t)

(12.9)

k¯ m−2 X,

A) = Pi {ξε (t) ∈ A,

k¯ m−2 τε

> t}

= Pi {k¯ m−2 τε ≤ t, ηε (k¯ m−2 τε ) = j}.

(12.10) (12.11)

288

12 Ergodic theorems for perturbed MARP

12.1.1.2 Perturbation Conditions for Embedded Alternating Regenerative Processes with Totally Removed Virtual Transitions. The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R, and k¯ m−2 M, assumed to be satisfied for the multialternating regenerative processes (ξε (t), ηε (t)), are also satisfied for the shifted multi-alternating regenerative processes (ξε(1) (t), ηε(1) (t)). The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R take for the alternating regenerative processes (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)) the following forms: k¯ m−2 G1 :(a) k¯ m−2 pε,i j > 0, k¯ m−2 Fε,i (0) = 0, ε

ε ∈ (0, 1] or k¯ m−2 pε,i j = 0, ε ∈ (0, 1], for i, j ∈ ∈ (0, 1], for i ∈ k¯ m−2 X.

k¯ m−2 H1 : k¯ m−2 p1,i j , k¯ m−2 p1, ji k¯ m−2 I1 : k¯ m−2 pε,i j

→

> 0, for i, j ∈

k¯ m−2 p0,i j

k¯ m−2 X,

as ε → 0, for i, j ∈

k¯ m−2 X,

(b)

i  j. k¯ m−2 X.

The functions k¯ m−2 p ·,i j , j ∈ k¯ m−2 Y1,i, i ∈ X belong to the complete family of asymptotically comparable functions H appearing in condition IH,1 .

k¯ m−2 IH,1 :

k¯ m−2 J1 :(a) k¯ m−2 Fε,i j (· k¯ m−2 uε,i ) ⇒ k¯ m−2 F0,i j (·) as ε → 0, for j ∈ k¯ m−2 Y1,i , i ∈ k¯ m−2 X, (b) k¯ m−2 F0,i j (·) is a non-arithmetic distribution function without singular com-

ponent, for j ∈ i ∈ k¯ m−2 X.

k¯ m−2 Y1,i,

i∈

k¯ m−2 X

(c)

k¯ m−2 uε,i

∈ (0, ∞), ε ∈ (0, 1], for

And, equivalently, ◦ k¯ m−2 J1 :

k¯ m−2 φε,i j (s/ k¯ m−2 uε,i ) → k¯ m−2 φ0,i j (s) as ε → 0, for s ≥ 0 and j ∈ Y k¯ m−2 1,i , i ∈ k¯ m−2 X, (b) k¯ m−2 φ0,i j (·) is the Laplace transform of non-arithmetic distribution function without singular component k¯ m−2 F0,i j (·), for j ∈ k¯ m−2 Y1,i, i ∈ k¯ m−2 X, (c) k¯ m−2 uε,i ∈ (0, ∞), ε ∈ (0, 1], for i ∈ k¯ m−2 X.

(a)

k¯ m−2 K1 :(a) k¯ m−2 fε,i j < ∞, for j ∈ k¯ m−2 Y1,i, k¯ m−2 fε,i j / k¯ m−2 uε,i → k¯ m−2 f0,i j < ∞ as

i ∈ k¯ m−2 X and ε ∈ (0, 1], (b) ε → 0, for j ∈ k¯ m−2 Y1,i, i ∈ k¯ m−2 X, (c) k¯ m−2 f0,i j is the first moment of the distributions function k¯ m−2 F0,i j (·), for j ∈ k¯ m−2 Y1,i, i ∈ k¯ m−2 X.

k¯ m−2 L1 : k¯ m−2 uε,i

→

k¯ m−2 u0,i

∈ (0, ∞] as ε → 0, for i ∈

k¯ m−2 X.

The functions k¯ m−2 u ·,i, i ∈ k¯ m−2 X belong to a complete family of asymptotically comparable functions H (appearing in condition LH,1 ).

k¯ m−2 LH,1 :

k¯ m−2 R:

There exist functions k¯ m−2 q0,i (t, A), t ≥ 0, A ∈ BZ , i ∈ k¯ m−2 X, which belong to the class P[BZ ], a class of sets Γ ⊆ BZ , and Borel sets U[k¯ m−2 q ·,i (· k¯ m−2 u ·,i , A)], A ∈ Γ, i ∈ k¯ m−2 X such that: (a) the function k¯ m−2 q0,i (t, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − k¯ m−2 F0,i (t), t ∈ R+ , us

for i ∈ k¯ m−2 X; (b) the functions k¯ m−2 qε,i (· k¯ m−2 uε,i, A) −→ k¯ m−2 q0,i (·, A) as ε → 0, for s ∈ U[k¯ m−2 q ·,i (· k¯ m−2 u ·,i, A)], A ∈ Γ, i ∈ k¯ m−2 X; (c) m(U¯ [k¯ m−2 q ·,i (· k¯ m−2 u ·,i , A)]) = 0, for A ∈ Γ, i ∈ k¯ m−2 X; (d) the function k¯ m−2 q0,i (·, A) is continuous almost everywhere with respect to the Lebesgue measure m(·) on B+ , for A ∈ Γ, i ∈ k¯ m−2 X.

12.1 Embedded ARP

289

The space k¯ m−2 X = {k m−1, k m } is a two-state set. The normalisation functions k¯ m−2 uε,i, i ∈ k¯ m−2 X are given by the relations (11.10)–(11.12) and take the following forms: k¯ m−2 uε,i

=

m−3 

−1 k¯r p¯ε,ii uε,i .

(12.12)

r=0

The distribution functions k¯ m−2 F0,i (·), i ∈ k¯ m−2 X appearing in the condition k¯ m−2 R are defined by the following relation:

(12.13) k¯ m−2 F0,i j (·) k¯ m−2 p0,i j k¯ m−2 F0,i (·) = j ∈ k¯

m−2

X

The following lemma, which is a direct corollary of Theorem 11.1, takes place. Lemma 12.1 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R, and k¯ m−2 M be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Then: (i) The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , and R are satisfied for the shifted alternating regenerative processes (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)) in the form of conditions k¯ m−2 G1 , k¯ m−2 H1 , k¯ m−2 I1 , k¯ m−2 IH,1 , k¯ m−2 L1 , k¯ m−2 LH,1 , k¯ m−2 J1 (k¯ m−2 J◦1 ), k¯ m−2 K1 , k¯ m−2 R. (ii) The quantities and sets appearing in the above conditions can be calculated using the recurrent algorithms described in Lemmas 10.4 - 10.10, 10.19 - 10.24, and Theorem 11.1. (iii) The normalising functions u · = k¯ m−2 u ·,i, i = k m−1, k m are given by the relation (12.12). In what follows, the following condition plays an important role: k¯ m−2 Sβ :

k¯ m−2 p ε, k m−1 , k m

k¯ m−2 p ε, k m , k m−1

→β=

k¯ m−2 β

∈ [0, ∞] as ε → 0.

According to Lemma 12.1, the conditions G1 , H1 , and I1 , IH,1 imply that the conditions k¯ m−2 G1 , k¯ m−2 H1 , and k¯ m−2 I1 , k¯ m−2 IH,1 are satisfied and, thus, the condition k¯ m−2 Sβ is satisfied. Note that the parameter k¯ m−2 β can be calculated using Lemmas 8.21 –8.91 . Let us also introduce conditions: k¯ m−2 T4 : (a) k¯ m−2 p0,k m,k m−1

> 0 or (b) k¯ m−2 p0,km−1,km > 0.

¯ k¯ m−2 T4 : k¯ m−2 p0,k m,k m−1 , k¯ m−2 p0,k m−1,k m

= 0.

The condition k¯ m−2 T4 corresponds to regularly and semi-regularly perturbed em-

bedded alternating regenerative processes (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)). The condition k¯ m−2 T¯ 4 would correspond to singularly perturbed embedded alter-

nating regenerative processes (k¯ m−2 ξε(1) (t),

(1) k¯ m−2 ηε (t)).

290

12 Ergodic theorems for perturbed MARP

However, according to Lemma 11.2, the conditions G1 , H1 , and I1 , IH,1 imply that condition k¯ m−2 T4 is satisfied for the embedded alternating regenerative processes (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)), while the condition k¯ m−2 T¯ 4 cannot be satisfied for these processes. Note that the probabilities k¯ m−2 p0,km−1,km , k¯ m−2 p0,km,km−1 can be calculated using Lemmas 8.21 –8.91 . In the case, where the condition k¯ m−2 T4 is satisfied, k¯ m−2 β

=

k¯ m−2 p0,k m−1,k m k¯ m−2 p0,k m,k m−1

.

(12.14)

Note that the condition k¯ m−2 T4 (a) is satisfied if and only if the condition k¯ m−2 T4 is satisfied and k¯ m−2 β ∈ [0, ∞). Similarly, the condition k¯ m−2 T4 (b) is satisfied if and only if the condition k¯ m−2 T4 is satisfied and k¯ m−2 β ∈ (0, ∞]. 12.1.1.3 Perturbation Conditions for Characteristics Related to Transition Period of Embedded Alternating Regenerative Processes with Totally Removed Virtual Transitions. Let us also analyse the conditions associated with the transition period [0, k¯ m−2 τε ). Let us use the notation, Dm−1 =

k¯ m−2 X

¯ m−1 = {k1, . . . , k m−2 }. = {k m−1, k m }, D

(12.15)

The stopping time k¯ m−2 τε is connected with the first hitting time τε,Dm−1 in the domain Dm−1 (for the semi-Markov process ηε (t)) by the following relation: k¯ m−2 τε

= I(ηε (0) ∈

¯ k¯ m−2 X) τε,D m−1 .

(12.16)

Let us introduce the distribution functions, for k n ∈ X,  k¯ m−2 Fε,k n (·)

= Pkn {k¯ m−2 τε ≤ ·}  =

I(· ≥ 0) for k n ∈ Dm−1, ¯ m−1, Pk n {τε,D m−1 ≤ ·} for k n ∈ D

(12.17)

and the hitting probabilities, for k n ∈ X, j ∈ Dm−1 , k¯ m−2 pε,k n, j

= Pkn {ηε (k¯ m−2 τε ) = j}  =

I(k n = j) for k n ∈ Dm−1, ¯ m−1 . Pk n {ηε (τε,D m−1 ) = j} for k n ∈ D

(12.18)

From Lemma 10.141 and Theorems 10.21 –10.41 it follows that the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , and k¯ m−2 M imply that the following relations hold, ¯ m−1, j ∈ Dm−1 , for k n ∈ D Pk n {ηε (τε,D m−1 ) = j} → P0,D m−1,k n j as ε → 0,

(12.19)

12.1 Embedded ARP

291

and Pk n {τε,D m−1 ≤ · uˇ ε,k n , ηε (τε,D m−1 ) = j}

⇒ F0,Dm−1,kn j (·)P0,Dm−1,kn j as ε → 0,

(12.20)

¯ m−1 are given by the relation where: (a) the normalisation functions uˇ ·,kn , k n ∈ D ¯ m−1 are distribution functions concentrated on (10.115)1 ; (b) F0,Dm−1,kn j (·), k n ∈ D the interval [0, ∞), with Laplace transforms given by the relations (10.125)1 and (10.131)1 ; (c)the hitting probabilities P0,Dm−1,kn j satisfy the relations P0,Dm−1,kn j ≥ ¯ m−1 , and are given by the relation 0, j ∈ Dm−1, j ∈Dm−1 P0,Dm−1,kn j = 1, for k n ∈ D (10.85)1 . ¯ m−1, j ∈ Dm−1 , It is natural to denote, for k n ∈ D  for k n ∈ Dm−1, j ∈ Dm−1, 0,kn j (·) = I(· ≥ 0) (12.21) ¯k m−2 F ¯ m−1, j ∈ Dm−1, F0,Dm−1,kn j (·) for k n ∈ D 

and k¯ m−2 p0,k n j

=

I(k n = j) for k n ∈ Dm−1, j ∈ Dm−1, ¯ m−1, j ∈ Dm−1 . P0,Dm−1,kn j for k n ∈ D

(12.22)

The relation (12.20) implies that k¯ m−2 pε,k n j

= Pkn {ηε (k¯ m−2 τε ) = j} →

k¯ m−2 p0,k n j

as ε → 0,

(12.23)

and  k¯ m−2 Q ε,k n j (· uˇ ε,k n )

= Pkn {k¯ m−2 τε ≤ · uˇε,kn , ηε (k¯ m−2 τε ) = j} ⇒ k¯ Q 0,kn j (·) as ε → 0, m−2

(12.24)

where  I(· ≥ 0)I(k n = j) for k n ∈ Dm−1, j ∈ Dm−1,  k¯ m−2 Q 0,k n j (·) = 0,kn j (·)k¯ p0,kn j for k n ∈ D ¯ m−1, j ∈ Dm−1 . ¯k m−2 F m−2

(12.25)

The relation (12.24) implies that the following condition related to the transition period [0, k¯ m−2 τε ) is satisfied: k¯ m−2 P2 :

k¯ m−2 pε,k n j → k¯ m−2 p0,k n j as ε → 0, for n = 1, . . . , m, j ∈ Dm−1 , (b)   Q k¯ m−2 ε,k n j (· uˇ ε,k n ) ⇒ k¯ m−2 Q 0,k n j (·) as ε → 0, for n = 1, . . . , m, j ∈ Dm−1 .

(a)

It is important that according to the relation (10.88)1 , uˇε,km−2 =

m−3 

−1 k¯l p¯ε,k m−2 k m−2 uε,k m−2

(12.26)

l=0

and, according to the relation (10.118)1 given in Lemma 10.151 , for any 1 ≤ n ≤ m − 2,

292

12 Ergodic theorems for perturbed MARP

uˇε,kn → uˇε,km−2

ˇ 0,kn,km−2 k¯ m−2 w

∈ [0, ∞) as ε → 0.

(12.27)

The relations (11.12) and (12.26) imply that, for i = k m−1, k m , uˇε,km−2 =

k¯ m−3 u˜ ε,k m−2

and k¯ m−2 uε,i =

k¯ m−3 u˜ ε,i .

(12.28)

Thus, the condition k¯ m−2 M implies that, for i = k m−1, k m , uˇε,km−2 = k¯ m−2 uε,i

k¯ m−3 u˜ ε,k m−3 k¯ m−3 u˜ ε,i

→

k¯ m−3 w0,k m−2,i

∈ [0, ∞) as ε → 0.

(12.29)

Lemma 12.2 Let the conditions G1 , H1 , I1 , IH,1 , J1 (J◦1 ), K1 , and k¯ m−2 M be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)) with the transition periods [0, k¯ m−2 τε ). Then: (i) The condition k¯ m−2 P2 is satisfied. (ii) The quantities and sets appearing in the condition k¯ m−2 P2 can be calculated using the recurrent algorithms described in Lemmas 10.131 –10.141 and Theorems 10.21 –10.41 . (iii) The normalisation functions uˇ ·,kn , n = 1, . . . , m − 2, are given by the relation (10.115)1 . (iv) Asymptotic relations (12.27) and (12.29) hold. 12.1.1.4 Compressed in Time Embedded Alternating Regenerative Processes with Totally Removed Virtual Transitions. Due to the conditions IH,1 and LH,1 , the normalisation functions uˇε,kn , 1 ≤ n ≤ m − 2 and k¯ m−2 uε,i, i = k m−1, k m belong to the class of asymptotically comparable functions H. This allows us to introduce universal normalisation functions, uε =

k¯ m−2 uε

=

k¯ m−2 uε,k m−1

+

k¯ m−2 uε,k m .

(12.30)

Note that the conditions IH,1 and LH,1 imply that uε =

k¯ m−2 uε

→ u0 =

k¯ m−2 u0

∈ (0, ∞] as ε → 0.

(12.31)

The conditions G1 , H1 , and IH,1 , LH,1 imply that the following condition is satisfied: k¯ m−2 Xγ :

k¯ m−2 u ε, k m k¯ m−2 u ε, k m−1

→γ=

k¯ m−2 γ

∈ [0, ∞] as ε → 0.

From the conditions IH,2 , LH,1 it follows that the limit (in the asymptotic relations given in the condition k¯ m−2 Xγ ) exists. Moreover, Lemmas 8.21 –8.91 allow us to calculate these limits. The condition k¯ m−2 Xγ simply separates the cases, where these limits take different values in the interval [0, ∞]. The condition k¯ m−2 Xγ implies that, for i = k m−1, k m ,

12.1 Embedded ARP

293

lim

k¯ m−2 uε,i

ε→0 k¯ u m−2 ε

 =

(1 + (1 +

−1 k¯ m−2 γ) −1 −1 k¯ m−2 γ )

for i = k m−1, for i = k m .

(12.32)

It is worth noting that the limits in the relations (12.27), (12.29), (12.32), and in the condition k¯ m−2 Xγ can be calculated using Lemmas 8.21 –8.91 given in Chap. 81 . Consider the following multi-alternating regenerative process compressed in time, (ξε,uε (t), ηε,uε (t)) = (ξε (tuε ), ηε (tuε )), t ≥ 0,

(12.33)

where the time compression factor uε = k¯ m−2 uε is given by the relation (12.30). The process (ξε,uε (t), ηε,uε (t)) can be considered as a multi-alternating regenerative process with a transition period [0, k¯ m−2 τε,uε ), where k¯ m−2 τε,uε

=

k¯ m−2 τε

uε

.

(12.34)

Let us also consider the corresponding shifted multi-alternating regenerative process compressed in time, (1) (1) (ξε,u (t), ηε,u (t)) = (ξε,uε (k¯ m−2 τε,uε + t), ηε,uε (k¯ m−2 τε,uε + t)), t ≥ 0. ε ε

(12.35)

Obviously, the condition k¯ m−2 W is satisfied for these multi-alternating regenerative processes. (1) (1) (t), k¯ m−2 ηε,u (t)) be the alternating regenerative process (with Let also (k¯ m−2 ξε,u ε ε the reduced phase space k¯ m−2 X = {k m−1, k m } of the modulating semi-Markov pro-

(1) (t)) built using the recurrent algorithm for time–space aggregation of cess k¯ m−2 ηε,u ε regeneration times (described in Sect. 11.2) applied to the multi-alternating regener(1) (1) (t), ηε,u (t)). ative process (ξε,u ε ε The following relation, which is equivalent to the relation (12.5), takes place, for i ∈ k¯ m−2 X, t ≥ 0, A ∈ BZ and ε ∈ (0, 1], (1) (1) (1) (t, A) = Pi {ξε,u (t) ∈ A} = Pi {k¯ m−2 ξε,u (t) ∈ A} Pε,u ε ε ε ,i (1) = Pε,i (tuε, A) = Pi {ξε(1) (tuε ) ∈ A}

= Pi {k¯ m−2 ξε(1) (tuε ) ∈ A}.

(12.36)

The relations (12.6)–(12.11), given for the multi-alternating regenerative process (ξε (t), ηε (t)) and the alternating regenerative process (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)) in Sect. 12.1.1.1, can be re-written for the multi-alternating regenerative processes compressed in time with factor uε . (1) For each A ∈ BZ , the probabilities Pε,u (t, A), i ∈ k¯ m−2 X are measurable funcε ,i tions of t ≥ 0 and the only bounded solution of the following system of renewal type equations (which is an equivalent version of the system of renewal type equations (12.6)),

294

12 Ergodic theorems for perturbed MARP

(1) Pε,u (t, A) = k¯ m−2 qε,uε ,i (t, A) ε ,i

∫ t (1) + Pε,u (t − s, A) k¯ m−2 Q ε,uε ,i j (ds), t ≥ 0, i ∈ ε, j j ∈ k¯

m−2

0

X

where, for A ∈ BZ, t ≥ 0, i, j ∈ k¯ m−2 qε,uε ,i (t,

k¯ m−2 X,

(12.37)

k¯ m−2 X,

A) = Pi {ξε (tuε ) ∈ A,

k¯ m−2 ζε,1

> tuε },

(12.38)

and k¯ m−2 Q ε,uε ,i j (t)

= Pi { k¯ m−2 ζε,1 ≤ tuε,

k¯ m−2 ηε (k¯ m−2 ζε,1 )

= j}.

(12.39)

Also, the probabilities Pε,uε ,i (t, A) = Pi {ξε,uε (t) ∈ A}, i ∈ X are, for every A ∈ BZ , measurable functions of t ≥ 0, which satisfy the following renewal type relations (which is an equivalent variant of the system of renewal equations (12.9)): Pε,uε ,i (t, A) = k¯ m−2 qε,uε ,i (t, A)

∫ t (1) Pε,u (t − s, A) k¯ m−2 Q ε,uε ,i j (ds), t ≥ 0, i ∈ X, + ε, j j ∈ k¯

m−2

where, for A ∈ BZ, t ≥ 0, i ∈ X, j ∈ k¯ m−2 qε,uε ,i (t,

and

(12.40)

0

X

 k¯ m−2 Q ε,uε ,i j (t)

k¯ m−2 X,

A) = Pi {ξε (tuε ) ∈ A,

k¯ m−2 τε

> tuε }

(12.41)

= Pi {k¯ m−2 τε ≤ tuε, ηε (k¯ m−2 τε ) = j}.

(12.42)

The transition characteristics of the compressed in time alternating regenerative (1) (1) process (k¯ m−2 ξε,u (t), k¯ m−2 ηε,u (t)) are connected with the transition characteristics ε ε of the alternating regenerative process (k¯ m−2 ξε(1) (t), lations, for ε ∈ (0, 1] and i, j ∈ k¯ m−2 X, k¯ m−2 Q ε,uε ,i j (t) k¯ m−2 pε,uε ,i j

=

k¯ m−2 ψε,uε ,i j (s) k¯ m−2 eε,uε ,i j

=

k¯ m−2 Q ε,i j (tuε ),

k¯ m−2 pε,i j

=

=

−1 = u−1 ε k¯ m−2 eε,i j = uε

k¯ m−2 qε,uε ,i (t,

A) =

k¯ m−2 qε,i (tuε ,

by the following re-

t ≥ 0,

k¯ m−2 Q ε,i j (∞), ∫ ∞

k¯ m−2 ψε,i j (s/uε )

(1) k¯ m−2 ηε (t))

= ∫

0

0 ∞

e−su/uε

k¯ m−2 Q ε,i j (du), s

≥ 0,

u k¯ m−2 Q ε,i j (du),

A), t ≥ 0, A ∈ BZ .

(12.43)

12.1 Embedded ARP

295

Let us prove that the conditions k¯ m−2 G1 k¯ m−2 H1 , k¯ m−2 I1 , k¯ m−2 J1 , k¯ m−2 K1 , and k¯ m−2 R ˆ 2 , and R ˆ 2 (formulated in Sect. 5.2) are satisfied imply that the conditions O2 , Pˆ 2 , Q for the shifted alternating regenerative processes (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)). The conditions k¯ m−2 G1 and k¯ m−2 H1 obviously imply that the condition O2 is satisfied. The conditions k¯ m−2 G1 , k¯ m−2 I1 , and k¯ m−2 J1 and Lemma 10.1 imply that the condition Pˆ 2 is satisfied. The conditions k¯ m−2 G1 , k¯ m−2 I1 , k¯ m−2 J1 , and k¯ m−2 K1 and Lemma 10.2 imply that ˆ 2 is satisfied. the condition Q ˆ 2 for the above shifted Finally, the condition k¯ m−2 R coincides with the condition R alternating regenerative processes. Let us now assume that the conditions k¯ m−2 I1 , k¯ m−2 J1 (k¯ m−2 J◦1 ), k¯ m−2 K1 , k¯ m−2 R, and k¯ m−2 Xγ are satisfied. The conditions k¯ m−2 I1 , k¯ m−2 J1 (k¯ m−2 J◦1 ), and k¯ m−2 Xγ imply that the following relations take place, for i, j ∈ k¯ m−2 X, k¯ m−2 pε,uε ,i j

=

k¯ m−2 pε,i j

→

as ε → 0,

k¯ m−2 p0,i j

(12.44)

and k¯ m−2 Q ε,uε ,i j (·)

= ⇒

where, for j ∈

k¯ m−2 Y1,i, i

∈

k¯ m−2 Q ε,i j (· k¯ m−2 uε,i (γ) k¯ m−2 Q 0,i j (·)

k¯ m−2 Y1,i, i

∈

)

as ε → 0,

(12.45)

for γ for γ for γ for γ

(12.46)

k¯ m−2 X,

⎧ ⎪ k¯ m−2 F0,i j (· (1 + γ)) k¯ m−2 p0,i j ⎪ ⎪ ⎨ ⎪ I(· ≥ 0) k¯ m−2 p0,i j (γ) −1 k¯ m−2 Q 0,i j (·) = ⎪ ⎪ k¯ m−2 F0,i j (· (1 + γ )) k¯ m−2 p0,i j ⎪ ⎪ I(· ≥ 0) ¯ p0,i j k m−2 ⎩ and, for j 

uε k¯ m−2 uε,i

∈ [0, ∞), i = k m−1, = ∞, i = k m−1, ∈ (0, ∞], i = k m, = 0, i = k m,

k¯ m−2 X, (γ) k¯ m−2 Q 0,i j (·)

= 0(·) ≡ 0.

(12.47)

The relations (12.44) and (12.45) are equivalent to the following convergence relation for the corresponding Laplace transforms, which takes place for i, j ∈ k¯ m−2 X, k¯ m−2 ψε,uε ,i j (s)

= →

where, for s ≥ 0, j ∈

k¯ m−2 Y1,i, i

∈

k¯ m−2 ψε,i j (s k¯ m−2 uε,i (γ) k¯ m−2 ψ0,i j (s) k¯ m−2 X,

uε ) k¯ m−2 uε,i

as ε → 0, for s ≥ 0,

(12.48)

296

12 Ergodic theorems for perturbed MARP (γ) k¯ m−2 ψ0,i j (s) −1 ⎧ ⎪ k¯ m−2 φ0,i j (s(1 + γ) ) k¯ m−2 p0,i j ⎪ ⎪ ⎨ ¯ p ⎪ = km−2 0,i j −1 −1 ⎪ k¯ m−2 φ0,i j (s(1 + γ ) ) k¯ m−2 p0,i j ⎪ ⎪ ⎪ ¯ p0,i j ⎩ km−2

and, for s ≥ 0, j 

k¯ m−2 Y1,i, i

∈

for γ for γ for γ for γ

∈ [0, ∞), i = k m−1, = ∞, i = k m−1, ∈ (0, ∞], i = k m, = 0, i = k m,

(12.49)

k¯ m−2 X, (γ) k¯ m−2 ψ0,i j (s)

= 1.

(12.50)

In what follows, we also use the following distribution functions, for i ∈

k¯ m−2 X,

(γ) k¯ m−2 F0,i (·)

=

 ⎧ ⎪ j ∈ k¯ X ⎪ m−2 ⎪ ⎨ ⎪

(γ) k¯ m−2 Q 0,i j (·)

⎪ I(· ≥ 0) ⎪ ⎪ ⎪ ⎩

for γ ∈ [0, ∞), i = k m−1, and γ ∈ (0, ∞], i = k m for γ = ∞, i = k m−1, and γ = 0, i = k m .

(12.51)

Note that the condition k¯ m−2 J1 implies that the distribution functions k¯ m−2 F0,i j (· (1+ γ)) are non-arithmetic, for j ∈ k¯ m−2 Y1,i, i = k m−1 , γ ∈ [0, ∞), and the distribution functions k¯ m−2 F0,i j (· (1 + γ −1 )) are non-arithmetic, for j ∈ k¯ m−2 Y1,i, i = k m , γ ∈ (0, ∞]. Taking into account the above remarks, we can conclude that the conditions ◦   ¯km−2 I1 , k¯ m−2 J1 (k¯ m−2 J1 ), and k¯ m−2 Xγ imply that the conditions P2 , P2 , or P2 are (1) satisfied for the alternating regenerative processes (k¯ m−2 ξε,u (t), ε (γ) k¯ m−2 Q 0,i j (·), i,

(1) k¯ m−2 ηε,uε (t)),

with

j ∈ k¯ m−2 X, respectively, in the corresponding limiting distributions the case where γ ∈ (0, ∞), γ = 0, or γ = ∞. The conditions k¯ m−2 I1 , k¯ m−2 J1 (k¯ m−2 J◦1 ), k¯ m−2 K1 , and k¯ m−2 Xγ imply that the following relation takes place, for i, j ∈ k¯ m−2 X, k¯ m−2 eε,uε ,i j

=

k¯ m−2 eε,i j

→ where, for j ∈

(γ) k¯ m−2 e0,i j

k¯ m−2 Y1,i, i

∈

uε (γ) k¯ m−2 e0,i j

=

k¯ m−2 eε,i j k¯ m−2 uε,i k¯ m−2 uε,i

uε

as ε → 0,

(12.52)

k¯ m−2 X,

−1 ⎧ ⎪ k¯ m−2 f0,i j (1 + γ) k¯ m−2 p0,i j ⎪ ⎪ ⎨0 ⎪ = −1 −1 ⎪ ⎪ k¯ m−2 f0,i j (1 + γ ) k¯ m−2 p0,i j ⎪ ⎪0 ⎩

for γ for γ for γ for γ

∈ [0, ∞), i = k m−1, = ∞, i = k m−1, ∈ (0, ∞], i = k m, = 0, i = k m,

(12.53)

12.1 Embedded ARP

and, for j 

297

k¯ m−2 Y1,i, i

∈

k¯ m−2 X, (γ) k¯ m−2 e0,i j

≡ 0.

(12.54)

Taking into account Lemma 10.2 and the above remarks, we can conclude that the conditions k¯ m−2 I2 , k¯ m−2 J2 (k¯ m−2 J◦2 ), k¯ m−2 K2 , and k¯ m−2 Xγ imply that the condition

(1) (1) Q2 is satisfied for the alternating regenerative processes (k¯ m−2 ξε,u (t), k¯ m−2 ηε,u (t)), ε ε (γ)

with the corresponding limiting expectations k¯ m−2 e0,i j , i, j ∈ k¯ m−2 X. In what follows, we also use the following quantities, for i ∈ k¯ m−2 X,

(γ) k¯ m−2 e0,i

i∈

=

 ⎧ ⎪ j ∈ k¯ X ⎪ m−2 ⎪ ⎨ ⎪

(γ) k¯ m−2 e0,i j (·)

⎪ 0 ⎪ ⎪ ⎪ ⎩

for γ ∈ [0, ∞), i = k m−1, and γ ∈ (0, ∞], i = k m for γ = ∞, i = k m−1, and γ = 0, i = k m .

(12.55)

Let us define sets, for A ∈ Γ (Γ is the class of sets appearing in condition R), for k¯ m−2 X, U[γ, k¯ m−2 q ·,u·,i (· , A)]

⎧ {s ≥ 0 : s(1 + γ) ∈ U[k¯ m−2 q ·,i (· k¯ m−2 u · ,i, A)]} ⎪ ⎪ ⎪ ⎪ for γ ∈ [0, ∞), i = k m−1, ⎪ ⎪ ⎪ ⎪ [0, ∞) ⎪ ⎪ ⎪ ⎨ ⎪ for γ = ∞, i = k m−1, = [0, ∞) ⎪ ⎪ ⎪ ⎪ for γ = 0, i = k m, ⎪ ⎪ ⎪ ⎪ {s ≥ 0 : s(1 + γ −1 ) ∈ U[k¯ m−2 q ·,i (· k¯ m−2 u · ,i, A)]} ⎪ ⎪ ⎪ ⎪ for γ ∈ (0, ∞], i = k m . ⎩

(12.56)

Obviously, m(U¯ [γ, k¯ m−2 q0,u·,i (· , A)]) = 0, for γ ∈ [0, ∞], A ∈ Γ, i ∈ k¯ m−2 X, since m(U¯ [k¯ m−2 q ·,i (· k¯ m−2 u · ,i, A)]) = 0, for A ∈ Γ, i ∈ k¯ m−2 X, by the condition k¯ m−2 R. The conditions k¯ m−2 R and Xγ imply that the following relation takes place, for s ∈ U[γ, k¯ m−2 q ·,u·,i (· , A)], γ ∈ [0, ∞], A ∈ Γ, i ∈ k¯ m−2 X, k¯ m−2 qε,uε ,i (·,

A) =

k¯ m−2 qε,i (· uε ,

A) us uε = k¯ m−2 qε,i (· uε,i , A) −→ uε,i

where, for s ≥ 0, A ∈ BZ, i ∈

k¯ m−2 X,

(γ) k¯ m−2 q0,i (·,

A) as ε → 0,

(12.57)

298

12 Ergodic theorems for perturbed MARP

(γ) k¯ m−2 q0,i (s,

⎧ ⎪ ⎪ ⎪ ⎨ ⎪

k¯ m−2 q0,i (s(1

+ γ), A) 0(s) = 0 A) = −1 ⎪ k¯ m−2 q0,i (s(1 + γ ), A) ⎪ ⎪ ⎪ 0(s) = 0 ⎩

The relation (12.58) and the condition (γ) k¯ m−2 q0,i (s, Z), s

for γ for γ for γ for γ

∈ [0, ∞), i = k m−1, = ∞, i = k m−1, ∈ (0, ∞], i = k m, = 0, i = k m

k¯ m−2 R

(12.58)

(a) imply that the function (γ)

≥ 0 is consistent with the tail probability function 1− k¯ m−2 F0,i (s), s ≥ 0 for γ ∈ [0, ∞), i = k m−1 and γ ∈ (0, ∞], i = k m . This consistency relation also holds for γ = ∞, i = k m−1 and γ = 0, i = k m , since the function 0(s) = 0, s ≥ 0 is consistent with the tail probability function 1 − I(s ≥ 0), s ≥ 0. Also, the relation (12.58) and the condition k¯ m−2 R (d) imply that the function (γ) k¯ m−2 q0,i (·,

A) is continuous almost everywhere with respect to the Lebesgue measure m(·) on B+ , for γ ∈ [0, ∞], A ∈ Γ, i ∈ k¯ m−2 X. Therefore, the conditions k¯ m−2 I1 , k¯ m−2 J1 (k¯ m−2 J◦1 ), k¯ m−2 K1 , k¯ m−2 R, and k¯ m−2 Xγ imply that the condition R2 is satisfied for the alternating regenerative processes (γ) (1) (1) (k¯ m−2 ξε,u (t), k¯ m−2 ηε,u (t)), with the corresponding limiting functions k¯ m−2 q0,i (s, A), ε ε for γ ∈ [0, ∞], i ∈ k¯ m−2 X, A ∈ BZ . It is also useful to note that, according to the relations (12.53) and (12.58), the corresponding limiting stationary probabilities, ∫ ∞ 1 (γ) (γ) k¯ m−2 π0,i (A) = k¯ m−2 q0,i (s, A)m(ds) (γ) k¯ m−2 e0,i 0 ∫ ∞ 1 = (12.59) k¯ m−2 q0,i (s, A)m(ds) = k¯ m−2 π0,i (A), k¯ m−2 e0,i 0 do not depend on the parameter γ = k¯ m−2 γ in the cases γ = k m−1, A ∈ BZ and γ = k¯ m−2 γ ∈ (0, ∞], i = k m, A ∈ BZ . The following lemma summarises the above remarks.

k¯ m−2 γ

∈ [0, ∞), i =

Lemma 12.3 Let the conditions G1 , H1 , I1 , IH,1 , J1 (J◦1 ), K1 , R, and k¯ m−2 M be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)) and, thus, the conditions k¯ m−2 G1 , k¯ m−2 H1 , k¯ m−2 I1 , k¯ m−2 IH,1 , k¯ m−2 J1 (k¯ m−2 J◦1 ), k¯ m−2 K1 , k¯ m−2 R, k¯ m−2 T4 , and k¯ m−2 Xγ (for some γ ∈ [0, ∞]) are satisfied for the shifted alternating regenerative processes (k¯ m−2 ξε(1) (t),

(1) k¯ m−2 ηε (t)). Then the following conditions are satisfied for the (1) (1) (t), k¯ m−2 ηε,u (t)), alternating regenerative processes (k¯ m−2 ξε,u ε ε

compressed in time with the time compression factor u · = k¯ m−2 u · given by the relation (12.30): (i) If k¯ m−2 γ ∈ (0, ∞), then the conditions O2 , P2 , Q2 , and R2 hold. (ii) If k¯ m−2 γ = 0, then the conditions O2 , P2 , Q2 , and R2 hold. (iii) If k¯ m−2 γ = ∞, then the conditions O2 , P2, Q2 , and R2 hold. (iv) The asymptotic relations (12.44), (12.45), (12.48), (12.52), and (12.57) play the roles of asymptotic relations appearing in the above conditions. The correspond-

12.1 Embedded ARP

299

ing quantities and sets appearing in these conditions and related relations are given by the relations (12.43)–(12.59). Remark 12.1 Lemma 12.3 is, in fact, a variant of Lemma 5.1, applied to the alternating regenerative processes (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)) and the compressed in time

(1) (1) (t), k¯ m−2 ηε,u (t)). It follows from the alternating regenerative processes (k¯ m−2 ξε,u ε ε ˆ 2, R ˆ ˆ 2 are satisfied for above remarks, according to which the conditions O2 , P2 , Q (1) (1) the alternating regenerative processes (k¯ m−2 ξε (t), k¯ m−2 ηε (t)).

Remark 12.2 Lemmas 5.1 and 12.3 make it possible to apply Theorems 4.2– (1) (1) 4.4, 5.1, 5.2 to the alternating regenerative processes (k¯ m−2 ξε,u (t), k¯ m−2 ηε,u (t)) ε ε if the conditions G1 , H1 , I1 , IH,1 , J1 (J◦1 ), K1 , R, and k¯ m−2 M are satisfied. In this case, the conditions k¯ m−2 T4 , k¯ m−2 Xγ , and k¯ m−2 Sβ are also satisfied. Theorem 4.2 can be applied, if γ ∈ (0, ∞) and β ∈ (0, ∞), Theorem 4.3 can be applied, if γ ∈ (0, ∞) and β = 0, Theorem 4.4 can be applied, if γ ∈ (0, ∞) and β = ∞, Theorem 5.1 can be applied, if γ = 0 and β ∈ [0, ∞), and Theorem 5.2 can be applied, if γ = ∞ and β ∈ (0, ∞]. The cases, where the vector parameter (γ, β) takes value (0, ∞) or (∞, 0), are excluded, since Theorems 5.1 and 5.2 do not work in these cases for the reasons stated in Remark 5.5. 12.1.1.5 Perturbation Conditions for Characteristics Related to Transition Periods of Compressed in Time Embedded Alternating Regenerative Processes with Totally Removed Virtual Transitions. Let us also consider the conditions associated with the compressed transition period [0, k¯ m−2 τε,uε ). The conditions IH,1 , LH,1 and the relations (12.27), (12.29) imply that the following asymptotic relation holds, for 1 ≤ n ≤ m − 2, uˇε,kn → k¯ m−2 uε

 kn k¯ m−2 w

∈ [0, ∞) as ε → 0,

(12.60)

where

 kn k¯ m−2 w

=

−1 −1 ⎧ ˇ 0,kn,km−2 · (k¯ m−3 w0,k + k¯ m−3 w0,k )−1 ⎪ k¯ m−2 w ⎪ m−2,k m−1 m−2,k m ⎪ ⎨ ⎪ if ¯ w0,k ,k · ¯ w0,k ,k > 0,

⎪ 0 ⎪ ⎪ ⎪ ⎩

k m−3

m−2

m−1

k m−3

if k¯ m−3 w0,km−2,km−1 ·

m−2

m

k¯ m−3 w0,k m−2,k m

(12.61)

= 0.

The limits k¯ m−2 w kn , 1 ≤ n ≤ m − 2 can be calculated using Lemmas 8.21 –8.91 . The relation (12.60) and the conditions k¯ m−2 P2 and k¯ m−2 Xγ imply that the following relations hold, for n = 1, . . . , m, j = k m−1, k m , k¯ m−2 pε,k n j

and

→

k¯ m−2 p0,k n j

as ε → 0,

(12.62)

300

12 Ergodic theorems for perturbed MARP

 k¯ m−2 Q ε,uε ,k n j (·)

= →

 k¯ m−2 Q ε,k n j (· uˇ ε,k n  k¯ m−2 Q 0,k n j (·)

uε ) uˇε,kn

as ε → 0,

(12.63)

where  k¯ m−2 Q 0,k n j (·) =

⎧ for k n ∈ Dm−1, j ∈ Dm−1, ⎪ ⎨ I(· ≥ 0)I(k n = j) ⎪ ¯ m−1, j ∈ Dm−1, w kn = 0, for k n ∈ D I(· ≥ 0) k¯ m−2 p0,kn j ⎪ ⎪ k¯ F0,kn j (· w −1 ) k¯ p0,kn j for k n ∈ D ¯ m−1, j ∈ Dm−1, w kn > 0. k n m−2 ⎩ m−2

(12.64)

These relations imply that the condition P 2 formulated in Sect. 10.1.2.5 is satisfied. The above remarks can be summarised in the following lemma. Lemma 12.4 Let the conditions G1 , H1 , I1 , IH,1 , J1 (J◦1 ), K1 , and k¯ m−2 M, k¯ m−2 Xγ be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Then: (i) The condition P 2 is satisfied for the compressed in time multi-alternating regenerative processes (ξε,uε (t), ηε,uε (t)) with the transition periods [0, k¯ m−2 τε,uε ) and the time compression factor u · = k¯ m−2 u · given by the relation (12.30). (ii) The asymptotic relations (12.62) and (12.63) play the role of asymptotic relation appearing in the condition P 2 . The corresponding quantities and sets appearing in this condition are given by the relations (12.60), (12.61), and (12.64).

12.1.2 Embedded Alternating Regenerative Processes Based on Partial Removing of Virtual Transitions 12.1.2.1 Embedded Alternating Regenerative Processes with Transition Period and Partially Removed Virtual Transitions. Let (ξε (t), ηε (t)) be, for every ε ∈ (0, 1] the multi-alternating regenerative process with the phase space Z × X defined in Sect. 10.1.1. We assume that the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , and R are satisfied for these processes. Also, let (ηε,n, κε,n ) be the corresponding Markov renewal process used to construct the modulating semi-Markov process ηε (t). According to the remarks made in Sect. 11.2.1.3, there exists a sequence of states k¯ n = k1, . . . , k m−2 in the phase space X = {1, . . . , m} of the modulating semiMarkov process ηε (t) such that the condition k¯ m−2 M2 is satisfied. Let, also, (k¯ m−2 ξε (t), k¯ m−2 ηε (t)) be the multi-alternating regenerative process with the reduced phase space of modulating semi-Markov process k¯ m−2 X = X \ {k1, . . . , k m−2 } = {k m−1, k m } built using the recurrent algorithm for time– space aggregation of regeneration times (described in Sect. 11.3) applied to the multi-alternating regenerative process (ξε (t), ηε (t)).

12.1 Embedded ARP

301

In this case, (k¯ m−2 ξε (t), k¯ m−2 ηε (t)) is an alternating regenerative process with a two-state modulating semi-Markov process k¯ m−2 ηε (t). The states k m−1, k m are the two most absorbing states in the phase space X of the initial modulating semi-Markov processes ηε (t). It is useful to note that these two states are not compared in absorption rates. Let us assume that the initial distribution of the random variable ηε = ηε (0) satisfies the condition: k¯ m−2 W: P {ηε (0)

∈

The condition

k¯ m−2 X}

k¯ m−2 W

= 1, for ε ∈ (0, 1].

implies that, for ε ∈ (0, 1], P{k¯ m−2 ηε (t) ∈ k¯ m−2 X, t ≥ 0} = 1.

(12.65)

Therefore, (k¯ m−2 ξε (t), k¯ m−2 ηε (t)) = (ξε (t), k¯ m−2 ηε (t)) is an alternating regenerative process with a phase space Z × k¯ m−2 X. Let us now consider the general case when the condition k¯ m−2 W is not satisfied. Recall the stopping moments, defined, for ε ∈ (0, 1], by the relations (12.2) and (12.3),  0 if ηε,0 ∈ k¯ m−2 X, (12.66) k¯ m−2 αε = min(n ≥ 1 : η ε,n ∈ k¯ m−2 X) if ηε,0  k¯ m−2 X, and k¯ m−2 τε =

k¯ m−2 αε



κε,n .

(12.67)

n=1

The process (ξε (t), ηε (t)) can be considered as a multi-alternating regenerative process with a transition period [0, k¯ m−2 τε ). Let us consider the shifted multi-alternating regenerative process, (ξε(1) (t), ηε(1) (t)) = (ξε (k¯ m−2 τε + t), ηε (k¯ m−2 τε + t)), t ≥ 0.

(12.68)

Obviously, the condition k¯ m−2 W is satisfied for these multi-alternating regenerative processes. Let also (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)) be the multi-alternating regenerative processes (with the reduced phase space k¯ m−2 X = {k m−1, k m } for the modulating semi-Markov

processes k¯ m−2 ηε(1) (t))) built using the recurrent algorithm for time–space aggregation of regeneration times (described in Sect. 11.3) applied to the multi-alternating regenerative processes (ξε(1) (t), ηε(1) (t)). According to Theorem 11.2, the following relation takes place, for i ∈ k¯ m−2 X, t ≥ 0, A ∈ BZ and ε ∈ (0, 1], (1) (t, A) = Pi {ξε(1) (t) ∈ A} = Pi {k¯ m−2 ξε(1) (t) ∈ A}. Pε,i

(12.69)

302

12 Ergodic theorems for perturbed MARP

(1) For every A ∈ BZ , the probabilities Pε,i (t, A), i ∈ k¯ m−2 X are measurable functions of t ≥ 0 and the only bounded solution for the following system of renewal type equations (which is an analogue of the system of renewal type equations (12.6)), (1) Pε,i (t, A) =

 k¯ m−2 qε,i (t,



+

j ∈ k¯

m−2

A) ∫ t

(1)  Pε, j (t − s, A) k¯ m−2 Q ε,i j (ds), t ≥ 0, i ∈

0

X

where, for A ∈ BZ, t ≥ 0, i, j ∈  k¯ m−2 qε,i (t,

and

 k¯ m−2 Q ε,i j (t)

k¯ m−2 X,

(12.70)

k¯ m−2 X,

A) = Pi {ξε (t) ∈ A,

 = Pi { k¯ m−2 ζε,1 ≤ t,

 k¯ m−2 ζε,1

> t},

  k¯ m−2 ηε (k¯ m−2 ζε,1 )

= j}.

(12.71) (12.72)

Also, the probabilities Pε,i (t, A) = Pi {ξε (t) ∈ A}, i ∈ X are, for every A ∈ BZ , measurable functions of t ≥ 0, which satisfy the following renewal type relations (which are analogues of the relations (12.9)–(12.11)), Pε,i (t, A) =

k¯ m−2 qε,i (t,



+

j ∈ k¯

m−2

X

A) ∫ t 0

(1)  Pε, j (t − s, A) k¯ m−2 Q ε,i j (ds), t ≥ 0, i ∈ X,

where, for A ∈ BZ, t ≥ 0, i ∈ X, j ∈ k¯ m−2 qε,i (t,

and

 k¯ m−2 Q ε,i j (t)

(12.73)

k¯ m−2 X,

A) = Pi {ξε (t) ∈ A,

k¯ m−2 τε

> t}

= Pi {k¯ m−2 τε ≤ t, ηε (k¯ m−2 τε ) = j}.

(12.74) (12.75)

12.1.2.2 Perturbation Conditions for Embedded Alternating Regenerative Processes with Partially Removed Virtual Transitions. The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R, and k¯ m−2 M assumed to be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)) are also satisfied for the shifted multi-alternating regenerative processes (ξε(1) (t), ηε(1) (t)). The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R take for the alternating regenerative processes (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)) the following forms:   k¯ m−2 G1 :(a) k¯ m−2 pε,i j > 0,  k¯ m−2 Fε,i (0) = 0, ε

 ε ∈ (0, 1] or k¯ m−2 pε,i j = 0, ε ∈ (0, 1], for i, j ∈ ∈ (0, 1], for i ∈ k¯ m−2 X.

   k¯ m−2 H1 : k¯ m−2 p1,i j , k¯ m−2 p1, ji   k¯ m−2 I1 : k¯ m−2 pε,i j

→

> 0, for i, j ∈

 k¯ m−2 p0,i j

k¯ m−2 X,

as ε → 0, for i, j ∈

i  j. k¯ m−2 X.

k¯ m−2 X,

(b)

12.1 Embedded ARP

303

 k¯ m−2 IH,1 :

 The functions k¯ m−2 p·,i j , j ∈ k¯ m−2 Y1,i , i ∈ X belong to the complete family of asymptotically comparable functions H appearing in condition IH,1 .

     k¯ m−2 J1 :(a) k¯ m−2 Fε,i j (· k¯ m−2 uε,i ) ⇒ k¯ m−2 F0,i j (·) as ε → 0, for j ∈ k¯ m−2 Y1,i, i ∈ k¯ m−2 X,  (·) is a non-arithmetic distribution function without singular (b) k¯ m−2 F0,i j   component, for j ∈ k¯ m−2 Y1,i , i ∈ k¯ m−2 X, (c) k¯ m−2 uε,i ∈ (0, ∞), ε ∈ (0, 1],

for i ∈

k¯ m−2 X.

And, equivalently, ◦    k¯ m−2 J1 : (a) k¯ m−2 φε,i j (s/ k¯ m−2 uε,i ) → k¯ m−2 φ0,i j (s) as ε → 0, for s ≥ 0 and j ∈   k¯ m−2 Y1,i , i ∈ k¯ m−2 X, (b) k¯ m−2 φ0,i j (·) is the Laplace transform of non-arithme (·), for j ∈ tic distributions function without singular component k¯ m−2 F0,i j   k¯ m−2 Y1,i, i ∈ k¯ m−2 X, (c) k¯ m−2 uε,i ∈ (0, ∞), ε ∈ (0, 1], for i ∈ k¯ m−2 X.   k¯ m−2 L1 : k¯ m−2 uε,i

→

 k¯ m−2 u0,i

∈ (0, ∞] as ε → 0, for i ∈

k¯ m−2 X.

 k¯ m−2 LH,1 :

The functions k¯ m−2 u ·,i, i ∈ k¯ m−2 X belong to the complete family of asymptotically comparable functions H appearing in condition LH,1 .

   k¯ m−2 K1 :(a) k¯ m−2 fε,i j < ∞, for j ∈ k¯ m−2 Y1,i, i ∈ k¯ m−2 X and ε ∈ (0, 1], (b)     k¯ m−2 fε,i j / k¯ m−2 uε,i → k¯ m−2 f0,i j < ∞ as ε → 0, for j ∈ k¯ m−2 Y1,i, i ∈ k¯ m−2 X,  is the first moment of the distributions function  (c) k¯ m−2 f0,i k¯ m−2 F0,i j (·), for j  j ∈ k¯ m−2 Y1,i, i ∈ k¯ m−2 X. k¯ m−2 R



 (t, A), t ≥ 0, A ∈ B , i ∈ : There exist functions k¯ m−2 q0,i Z k¯ m−2 X, which belong  (· to the class P[BZ ], a class of sets Γ ⊆ BZ , and Borel sets U[k¯ m−2 q ·,i k¯ m−2 u ·,i ,  A)], A ∈ Γ, i ∈ k¯ m−2 X such that: (a) the function k¯ m−2 q0,i (t, A), t ∈ R+, A ∈  (t), t ∈ R , BZ is consistent with the tail probability function 1 − k¯ m−2 F0,i + us

 (·   for i ∈ k¯ m−2 X; (b) the functions k¯ m−2 qε,i k¯ m−2 uε,i, A) −→ k¯ m−2 q0,i (·, A)   as ε → 0, for s ∈ U[k¯ m−2 q ·,i (· k¯ m−2 u ·,i, A)], A ∈ Γ, i ∈ k¯ m−2 X; (c)  (·  m(U¯ [k¯ m−2 q ·,i k¯ m−2 u ·,i , A)]) = 0, for A ∈ Γ, i ∈ k¯ m−2 X; (d) the function  k¯ m−2 q0,i (·, A) is continuous almost everywhere with respect to the Lebesgue measure m(·) on B+ , for A ∈ Γ, i ∈ k¯ m−2 X.

The space k¯ m−2 X = {k m−1, k m } is a two-state set. The normalisation functions k¯ m−2 u ·,i, i ∈k¯ m−2 X are given by the recurrent relations (11.19)–(11.21).   (·), i ∈ The distribution functions k¯ m−2 F0,i k¯ m−2 X appearing in the condition k¯ m−2 R are defined by the following relation:

   (12.76) k¯ m−2 F0,i (·) = k¯ m−2 F0,i j (·) k¯ m−2 p0,i j j ∈ k¯

m−2

X

The following lemma, which is a direct corollary of Theorem 11.2, takes place.

304

12 Ergodic theorems for perturbed MARP

Lemma 12.5 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R, and k¯ m−2 M be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Then: (i) The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , and R are satisfied for the shifted alternating regenerative processes (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)) in the form   , k¯ m−2 L1 , k¯ m−2 LH,1 , k¯ m−2 J1 (k¯ m−2 J1◦ ), of conditions k¯ m−2 G1 , k¯ m−2 H1 , k¯ m−2 I1 , k¯ m−2 IH,1   k¯ m−2 K1 , k¯ m−2 R . (ii) The quantities and sets appearing in the above conditions can be calculated using the recurrent algorithms described in Lemmas 10.11 - 10.18, 10.26 - 10.31, and Theorem 11.2. (iii) The normalising functions u · = k¯ m−2 u ·,i, i = k m−1, k m are given by the relations (11.19)–(11.21). In what follows, the following condition plays an important role: p  k¯ m−2 ε, k m−1 , k m k¯ m−2 Sβ : k¯ pε, k , k m−2

m

m−1

→ β =

k¯ m−2 β



∈ [0, ∞] as ε → 0.

The conditions G1 , H1 and I1 , IH,1 imply that conditions k¯ m−2 G1 , k¯ m−2 H1 and   k¯ m−2 I1 , k¯ m−2 IH,1 are satisfied and, thus, the condition k¯ m−2 Sβ is also satisfied.  Note that the parameter k¯ m−2 β can be calculated using Lemmas 8.21 –8.91 . Let us introduce conditions:   k¯ m−2 T4 : (a) k¯ m−2 p0,k m,k m−1

 > 0 or (b) k¯ m−2 p0,k >0 m−1,k m

and   ¯ k¯ m−2 T4 : k¯ m−2 p0,k m,k m−1 , k¯ m−2 p0,k m−1,k m

= 0.

Condition k¯ m−2 T4 corresponds to the case, where embedded alternating regener-

ative processes (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)) are regularly or semi-regularly perturbed.  Condition k¯ m−2 T¯ 4 corresponds to the case, where embedded alternating regener-

ative processes (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)) are singularly perturbed.  According to Lemma 11.3, the condition k¯ m−2 T4 as well as the condition k¯ m−2 T¯ 4 can be satisfied for the perturbed embedded alternating regenerative processes (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)). The conditions separating the above two cases are described in Remark 11.1.  , ¯ p can be calculated using Note that the probabilities k¯ m−2 p0,k m−1,k m k m−2 0,k m,k m−1 Lemmas 8.21 –8.91 . In the case, where the condition k¯ m−2 T4 is satisfied,  k¯ m−2 β =

 k¯ m−2 p0,k m−1,k m  k¯ m−2 p0,k m,k m−1

.

(12.77)

Note also that the condition k¯ m−2 T4 (a) is satisfied if and only if the condition   k¯ m−2 T4 is satisfied and k¯ m−2 β ∈ [0, ∞).

12.1 Embedded ARP

305

Similarly, the condition k¯ m−2 T4 (b) is satisfied if and only if the condition k¯ m−2 T4 is satisfied and k¯ m−2 β  ∈ (0, ∞]. 12.1.2.3 Perturbation Conditions for Characteristics Related to Transition Period of Embedded Alternating Regenerative Processes with Partially Removed Virtual Transitions. Let us also analyse the conditions associated with the transition period [0, k¯ m−2 τε ). Recall the notation, Dm−1 =

k¯ m−2 X

¯ m−1 = {k1, . . . , k m−2 }. = {k m−1, k m }, D

(12.78)

The stopping time k¯ m−2 τε is connected with the first hitting time τε,Dm−1 in the domain Dm−1 (for the semi-Markov process ηε (t)) by the following relation: k¯ m−2 τε

= I(ηε (0) ∈

¯ k¯ m−2 X)τε,D m−1 .

(12.79)

Let us introduce the distribution functions, for k n ∈ X,  k¯ m−2 Fε,k n (·)

= Pkn {k¯ m−2 τε ≤ ·}  =

I(· ≥ 0) for k n ∈ Dm−1, ¯ m−1, Pk n {τε,D m−1 ≤ ·} for k n ∈ D

(12.80)

and the hitting probabilities, for k n ∈ X, j ∈ Dm−1 , k¯ m−2 pε,k n, j

= Pkn {ηε (k¯ m−2 τε ) = j}  =

I(k n = j) for k n ∈ Dm−1, ¯ m−1 . Pk n {ηε (τε,D m−1 ) = j} for k n ∈ D

(12.81)

From Lemma 10.141 and Theorems 10.21 –10.41 it follows that the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , and k¯ m−2 M imply that the following relations hold, ¯ m−1, j ∈ Dm−1 , for k n ∈ D Pk n {ηε (τε,D m−1 ) = j} → P0,D m−1,k n j as ε → 0,

(12.82)

and Pk n {τε,D m−1 ≤ · uˇ ε,k n , ηε (τε,D m−1 ) = j}

⇒ F0,Dm−1,kn j (·)P0,Dm−1,kn j as ε → 0,

(12.83)

¯ m−1 are given by the relawhere: (a) the normalisation functions uˇ ·,kn , k n ∈ D ¯ m−1 are distribution functions concentrated tion (10.115)1 ; (b) F0,Dm−1,kn j (·), k n ∈ D on the interval [0, ∞), with Laplace transforms given by relations (10.125)1 and (10.131)1 ; (c) the hitting probabilities P0,Dm−1,kn j satisfy the relations P0,Dm−1,kn j ≥  ¯ m−1 , and are given by the relation 0, j ∈ Dm−1, j ∈Dm−1 P0,Dm−1,kn j = 1, for k n ∈ D (10.85)1 . ¯ m−1, j ∈ Dm−1 , It is natural to denote, for k n ∈ D

306

12 Ergodic theorems for perturbed MARP

 k¯ m−2 F0,k n j (·)

 =

I(· ≥ 0) for k n ∈ Dm−1, j ∈ Dm−1, ¯ m−1, j ∈ Dm−1, F0,Dm−1,kn j (·) for k n ∈ D 

and k¯ m−2 p0,k n j

=

I(k n = j) for k n ∈ Dm−1, j ∈ Dm−1, ¯ m−1, j ∈ Dm−1 . P0,Dm−1,kn j for k n ∈ D

(12.84)

(12.85)

The relation (12.83) implies that, k¯ m−2 pε,k n j

= Pkn {ηε (k¯ m−2 τε ) = j} →

k¯ m−2 p0,k n j

as ε → 0,

(12.86)

and  k¯ m−2 Q ε,k n j (· uˇ ε,k n )

= Pkn {k¯ m−2 τε ≤ · uˇε,kn , ηε (k¯ m−2 τε ) = j} ⇒ k¯ Q 0,kn j (·) as ε → 0, m−2

(12.87)

where  k¯ m−2 Q 0,k n j (·)

 =

I(· ≥ 0)I(k n = j) for k n ∈ Dm−1, j ∈ Dm−1,  ¯ k¯ m−2 F0,k n j (·)k¯ m−2 p0,k n j for k n ∈ Dm−1, j ∈ Dm−1 .

(12.88)

The relation (12.87) implies that the following condition related to the transition period [0, k¯ m−2 τε ) is satisfied: k¯ m−2 P2 :

k¯ m−2 pε,k n j → k¯ m−2 p0,k n j as ε → 0, for n = 1, . . . , m, j ∈ Dm−1 , (b)   Q k¯ m−2 ε,k n j (· uˇ ε,k n ) ⇒ k¯ m−2 Q 0,k n j (·) as ε → 0, for n = 1, . . . , m, j ∈ Dm−1 .

(a)

It is important that according to the relation (10.88)1 , uˇε,km−2 =

m−3 

−1 k¯l p¯ε,k m−2 k m−2 uε,k m−2 ,

(12.89)

l=0

and, according to the relation (10.118)1 given in Lemma 10.151 , for any 1 ≤ n ≤ m − 2, uˇε,kn → k¯ m−2 wˇ 0,kn,km−2 ∈ [0, ∞) as ε → 0. (12.90) uˇε,km−2 Recall the set k¯ m−3 K[k m−2 ] introduced in the relation (11.19). According to the remarks made in Sect. 11.4.2, either k¯ m−3 K[k m−2 ] = {k m−2, k m−1, k m }, or k¯ m−3 K[k m−2 ] = {k m−2, k m−1 }, or k¯ m−3 K[k m−2 ] = {k m−2, k m }. The relations (11.12), (11.20) and the relations (11.28), (11.29) given in Lemma 11.1, together with the relation (12.89), imply that, for i ∈ k¯ m−3 K[k m−2 ] ∩ k¯ m−2 X,  , (12.91) uˇε,km−2 = k¯ m−3 u˜ε,km−2 = k¯ m−2 u˜ε,k m−2 and

 k¯ m−2 uε,i

=

 k¯ m−2 u˜ ε,i

=

k¯ m−3 u˜ ε,i .

(12.92)

12.1 Embedded ARP

307

Thus, the condition k¯ m−2 M and the relation (11.31) given in Lemma 11.1 imply that, for i ∈ k¯ m−3 K[k m−2 ] ∩ k¯ m−2 X, uˇε,km−2  = k¯ m−2 uε,i

 k¯ m−2 u˜ ε,k m−2  k¯ m−2 u˜ ε,i

→

k¯ m−3 w0,k m−2,i

∈ [0, ∞) as ε → 0.

(12.93)

The following lemma takes place. Lemma 12.6 Let the conditions G1 , H1 , I1 , IH,1 , J1 (J◦1 ), K1 , and k¯ m−2 M be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)) with the transition periods [0, k¯ m−2 τε ). Then: (i) The condition k¯ m−2 P2 is satisfied. (ii) The quantities and sets appearing in the condition k¯ m−2 P2 can be calculated using the recurrent algorithms described in Lemmas 10.131 –10.141 and Theorems 10.21 –10.41 . (iii) The normalisation functions uˇ ·,kn , n = 1, . . . , m − 2, are given by the relation (9.115)1 . (iv) The asymptotic relations (12.90) and (12.93) hold. 12.1.2.4 Compressed in Time Embedded Alternating Regenerative Processes with Partially Removed Virtual Transitions. Due to the conditions IH,1 and LH,1 , the normalisation functions uˇ ·,kn , 1 ≤ n ≤ m − 2 and k¯ m−2 u ·,i, i = k m−1, k m , belong to the class of asymptotically comparable functions H. This allows us to introduce universal normalisation functions, uε =

 k¯ m−2 uε

=

 k¯ m−2 uε,k m−1

+

 k¯ m−2 uε,k m .

(12.94)

  Note that the conditions IH,1 and LH,1 imply that

uε =

 k¯ m−2 uε

→ u0 =

 k¯ m−2 u0

∈ (0, ∞] as ε → 0.

(12.95)

The conditions G1 , H1 , and IH,1 , LH,1 imply that the following condition is satisfied:  k¯ m−2 Xγ :

 k¯ m−2 u ε, k m  u ¯ k m−2 ε, k m−1

→ γ =

k¯ m−2 γ



∈ [0, ∞] as ε → 0.

From the conditions IH,2 , LH,1 it follows that the limit (in the asymptotic relation given in the condition k¯ m−2 Xγ  ) exists. Moreover, Lemmas 8.21 –8.91 allow us to calculate these limits. Condition k¯ m−2 Xγ  simply separates the cases, where these limits take different values in the interval [0, ∞]. The condition k¯ m−2 Xγ  implies that, for i = k m−1, k m ,  k¯ m−2 uε,i lim ε→0 k¯ u m−2 ε

 =

(1 + (1 +

 −1 k¯ m−2 γ ) −1 )−1 k¯ m−2 γ

for i = k m−1, for i = k m .

(12.96)

308

12 Ergodic theorems for perturbed MARP

It is worth noting that the limits in the relations (12.90), (12.93), (12.32), and in the condition k¯ m−2 Xγ  can be calculated using Lemmas 8.21 –8.91 given in Chap. 81 . Consider the following multi-alternating regenerative process compressed time, (ξε,uε (t), ηε,uε (t)) = (ξε (tuε ), ηε (tuε )), t ≥ 0,

(12.97)

where the time compression factor uε = k¯ m−2 uε is given by the relation (12.94). The process (ξε,uε (t), ηε,uε (t)) can be considered as a multi-alternating regenerative process with a transition period [0, k¯ m−2 τε,uε ), where  k¯ m−2 τε,uε

=

k¯ m−2 τε uε

.

(12.98)

Let us also consider the corresponding shifted multi-alternating regenerative process compressed in time, (1) (1)  (¯    (ξε,u  (t), ηε,u  (t)) = (ξε,u ε k m−2 τε,uε + t), ηε,uε (k¯ m−2 τε,uε + t)), t ≥ 0. ε

ε

(12.99)

Obviously, the condition k¯ m−2 W is satisfied for these multi-alternating regenerative processes. (1) (1) Let, also, (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)) be the alternating regenerative process (with ε ε the reduced phase space k¯ m−2 X = {k m−1, k m } of the modulating semi-Markov pro-

(1) (t))) built using the recurrent algorithm for time–space aggregation of cess k¯ m−2 ηε,u ε regeneration times (described in Sect. 11.3) applied to the multi-alternating regener(1) (1) ative process (ξε,u  (t), ηε,u  (t)). ε ε The following relation, which is equivalent to relation (12.69), takes place, for i ∈ k¯ m−2 X, t ≥ 0, A ∈ BZ and ε ∈ (0, 1], (1) (1) (1) Pε,u  ,i (t, A) = Pi {ξε,u  (t) ∈ A} = Pi {k¯ m−2 ξε,u  (t) ∈ A} ε

ε

ε

(1) = Pε,i (tuε , A) = Pi {ξε(1) (tuε ) ∈ A}

= Pi {k¯ m−2 ξε(1) (tuε ) ∈ A}.

(12.100)

The relations (12.70)–(12.75) given for the multi-alternating regenerative process (ξε (t), ηε (t)) and alternating regenerative process (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)) in Sect. 12.1.2.1 can be re-written for the multi-alternating regenerative processes compressed in time with factor uε . (1) For each A ∈ BZ , the probabilities Pε,u  (t, A), i ∈ k¯ m−2 X are the measurable ε ,i functions of t ≥ 0, which are the only bounded solution for the following system of renewal type equations (which is an equivalent variant of the system of renewal type equations (12.70)),

12.1 Embedded ARP

309

(1)  Pε,u  (t, A) = k¯ m−2 qε,u  ,i (t, A) ε ε ,i

∫ t (1)  + Pε,u  , j (t − s, A) k¯ m−2 Q ε,u  ,i j (ds), t ≥ 0, i ∈ ε j ∈ k¯

m−2

X

ε

0

where, for A ∈ BZ, t ≥ 0, i, j ∈  k¯ m−2 qε,uε ,i (t,

k¯ m−2 X,

(12.101)

k¯ m−2 X,

A) = Pi {ξε (tuε ) ∈ A,

 k¯ m−2 ζε,1

> tuε },

(12.102)

and  k¯ m−2 Q ε,uε ,i j (t)

 = Pi { k¯ m−2 ζε,1 ≤ tuε ,

  k¯ m−2 ηε (k¯ m−2 ζε,1 )

= j}.

(12.103)

Also, the probabilities Pε,uε ,i (t, A) = Pi {ξε,uε (t) ∈ A}, i ∈ X, are, for every A ∈ BZ , measurable functions of t ≥ 0, which satisfy the following renewal type relations (which is an equivalent variant of the renewal type relations (12.73), (12.74), and (12.75)), Pε,uε ,i (t, A) = k¯ m−2 qε,uε ,i (t, A)

∫ t (1)   ,i j (ds), t ≥ 0, i ∈ X, Pε,u +  , j (t − s, A) k¯ m−2 Q ε,u ε j ∈ k¯

m−2

X

0

ε

where, for A ∈ BZ, t ≥ 0, i ∈ X, j ∈  k¯ m−2 qε,uε ,i (t,

and

  k¯ m−2 Q ε,uε ,i j (t)

(12.104)

k¯ m−2 X,

A) = Pi {ξε (tuε ) ∈ A,

k¯ m−2 τε

> tuε }

(12.105)

= Pi {k¯ m−2 τε ≤ tuε , ηε (k¯ m−2 τε ) = j}.

(12.106)

The transition characteristics for the compressed in time alternating regenerative (1) (1) process (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)) are connected with the transition characteristics ε

ε

for the alternating regenerative process (k¯ m−2 ξε(1) (t), relations, for ε ∈ (0, 1] and i, j ∈ k¯ m−2 X,

(1) k¯ m−2 ηε (t))

   k¯ m−2 Q ε,uε ,i j (t) = k¯ m−2 Q ε,i j (tuε ), t ≥ 0,    k¯ m−2 pε,uε ,i j = k¯ m−2 pε,i j = k¯ m−2 Q ε,i j (∞), ∫ ∞      e−su/uε k¯ m−2 Q ε,i k¯ m−2 ψε,uε ,i j (s) = k¯ m−2 ψε,i j (s/uε ) = j (du), s 0 ∫ ∞  −1  −1  u k¯ m−2 Q ε,i k¯ m−2 eε,uε ,i j = uε k¯ m−2 eε,i j = uε j (du), 0    k¯ m−2 qε,uε ,i (t, A) = k¯ m−2 qε,i (tuε , A), t ≥ 0, A ∈ BZ .

by the following

≥ 0,

(12.107)

310

12 Ergodic theorems for perturbed MARP

Let us prove that the conditions

     k¯ m−2 G1 , k¯ m−2 H1 , k¯ m−2 I1 , k¯ m−2 J1 , k¯ m−2 K1 ,

and ˆ ˆ ˆ ¯ ˆ k¯ m−2 R imply that the conditions O2 , P2 , or P2 , Q2 , and R2 (formulated in Sects. 5.2 

and 8.2) are satisfied for the shifted alternating regenerative processes (k¯ m−2 ξε(1) (t), (1) k¯ m−2 ηε (t)).

The conditions k¯ m−2 G1 and k¯ m−2 H1 obviously imply that the condition O2 is satisfied. The conditions k¯ m−2 G1 , k¯ m−2 I1 , and k¯ m−2 J1 and Lemma 10.1 imply that the condi tion Pˆ 2 or Pˆ¯ 2 is satisfied, if, respectively, the condition k¯ m−2 T4 or k¯ m−2 T¯ 4 is satisfied.  The conditions k¯ m−2 G1 , k¯ m−2 I1 , k¯ m−2 J1 , and k¯ m−2 K1 and Lemma 10.2 imply that ˆ 2 is satisfied. the condition Q ˆ 2 for the above shifted Finally, the condition k¯ m−2 R coincides with the condition R alternating regenerative processes. Let us now assume that the conditions k¯ m−2 I1 , k¯ m−2 J1 (k¯ m−2 J1◦ ), k¯ m−2 K1 , k¯ m−2 R, and k¯ m−2 Xγ  are satisfied. The conditions k¯ m−2 I1 , k¯ m−2 J1 (k¯ m−2 J1◦ ), and k¯ m−2 Xγ  imply that the following relations take place, for i, j ∈ k¯ m−2 X,  k¯ m−2 pε,uε ,i j

=

 k¯ m−2 pε,i j

→

 k¯ m−2 p0,i j

as ε → 0,

(12.108)

and  k¯ m−2 Q ε,uε ,i j (·)

= ⇒

where, for j ∈

 k¯ m−2 Y1,i, i

∈

  k¯ m−2 Q ε,i j (· k¯ m−2 uε,i (γ ) k¯ m−2 Q 0,i j (·)

as ε → 0,

 k¯ m−2 Y1,i, i

∈

(12.109)

k¯ m−2 X,

   ⎧ ⎪ k¯ m−2 F0,i j (· (1 + γ )) k¯ m−2 p0,i j ⎪ ⎪ ⎪  ⎨ I(· ≥ 0) k¯ p0,i j ⎪ (γ) m−2  (· (1 + γ −1 ))  k¯ m−2 Q 0,i j (·) = ⎪ F ¯ k k¯ m−2 p0,i j ⎪ m−2 0,i j ⎪ ⎪  ⎪ I(· ≥ 0) k¯ m−2 p0,i j ⎩

and, for j 

uε  ) k¯ m−2 uε,i

for γ  for γ  for γ  for γ 

∈ [0, ∞), i = k m−1, = ∞, i = k m−1, ∈ (0, ∞], i = k m, = 0, i = k m, (12.110)

k¯ m−2 X, (γ) k¯ m−2 Q 0,i j (·)

= 0(·) ≡ 0.

(12.111)

The relations (12.108) and (12.109) are equivalent to the following convergence relation for the corresponding Laplace transforms, which takes place for i, j ∈ k¯ m−2 X,

12.1 Embedded ARP

311

 k¯ m−2 ψε,uε ,i j (s)

  k¯ m−2 ψε,i j (s k¯ m−2 uε,i

= →

where, for s ≥ 0, j ∈

(γ ) k¯ m−2 ψ0,i j (s)

 k¯ m−2 Y1,i, i

∈

uε  ) k¯ m−2 uε,i

as ε → 0, for s ≥ 0,

k¯ m−2 X,

  −1  ⎧ ⎪ k¯ m−2 φ0,i j (s(1 + γ ) ) k¯ m−2 p0,i j ⎪ ⎪ ⎨ k¯ p0,i j ⎪ (γ ) m−2  −1 )−1 )  k¯ m−2 ψ0,i j (s) = ⎪ k¯ m−2 φ0,i j (s(1 + γ k¯ m−2 p0,i j ⎪ ⎪ ⎪ ¯ p ⎩ km−2 0,i j

and, for s ≥ 0, j 

 k¯ m−2 Y1,i, i

∈

(12.112)

for γ  for γ  for γ  for γ 

∈ [0, ∞), i = k m−1, = ∞, i = k m−1, ∈ (0, ∞], i = k m, = 0, i = k m, (12.113)

k¯ m−2 X, (γ ) k¯ m−2 ψ0,i j (s)

= 1.

(12.114)

In what follows, we also use the following distribution functions, for i ∈

k¯ m−2 X,

(γ ) k¯ m−2 F0,i (·)

=

 ⎧ ⎪ j ∈ k¯ X ⎪ m−2 ⎪ ⎨ ⎪

(γ) k¯ m−2 Q 0,i j (·)

⎪ I(· ≥ 0) ⎪ ⎪ ⎪ ⎩

for γ  ∈ [0, ∞), i = k m−1, and γ  ∈ (0, ∞], i = k m for γ  = ∞, i = k m−1, and γ  = 0, i = k m .

(12.115)

 (· (1+ Note that the condition k¯ m−2 J1 implies that the distribution functions k¯ m−2 F0,i j   γ)) are non-arithmetic, for j ∈ k¯ m−2 Y1,i, i = k m−1 , γ ∈ [0, ∞), and the distribu  (· (1 + γ −1 )) are non-arithmetic, for j ∈ tion functions k¯ m−2 F0,i k¯ m−2 Y1,i, i = k m , j  γ ∈ (0, ∞]. Taking into account the above remarks, we can conclude that the conditions   ◦  ¯  , P¯ 2 , or P¯  holds ¯km−2 I1 , k¯ m−2 J1 (k¯ m−2 J1 ), and k¯ m−2 Xγ imply that the condition P 2 2 (1) for the alternating regenerative processes (k¯ m−2 ξε,u  (t), ε

(γ) k¯ m−2 Q 0,i j (·), i,

(1) k¯ m−2 ηε,uε (t)),

with limiting

distributions j ∈ k¯ m−2 X, respectively, in the case where γ  = 0, γ  ∈ (0, ∞), or γ  = ∞. The conditions k¯ m−2 I1 , k¯ m−2 J1 (k¯ m−2 J1◦ ), k¯ m−2 K1 , and k¯ m−2 Xγ  imply that the following relation takes place, for i, j ∈ k¯ m−2 X,  k¯ m−2 eε,uε ,i j

=

 k¯ m−2 eε,i j uε

→ where, for j ∈

 k¯ m−2 Y1,i, i

∈

k¯ m−2 X,

=

(γ ) k¯ m−2 e0,i j

  k¯ m−2 eε,i j k¯ m−2 uε,i  uε k¯ m−2 uε,i

as ε → 0,

(12.116)

312

12 Ergodic theorems for perturbed MARP

(γ ) k¯ m−2 e0,i j

and, for j 

  −1  ⎧ ⎪ k¯ m−2 f0,i j (1 + γ ) k¯ m−2 p0,i j ⎪ ⎪ ⎨0 ⎪ =  −1 )−1  ⎪ k¯ m−2 p0,i j k¯ m−2 f0,i j (1 + γ ⎪ ⎪ ⎪0 ⎩  k¯ m−2 Y1,i, i

∈

for γ  for γ  for γ  for γ 

∈ [0, ∞), i = k m−1, = ∞, i = k m−1, ∈ (0, ∞], i = k m, = 0, i = k m,

(12.117)

k¯ m−2 X, (γ ) k¯ m−2 e0,i j

≡ 0.

(12.118)

Taking into account the above remarks, we can conclude that the conditions k¯ m−2 I1 ,  ◦   k¯ m−2 J1 (k¯ m−2 J1 ), k¯ m−2 K1 , and k¯ m−2 Xγ imply that condition Q2 is satisfied for the al(1) (1) ternating regenerative processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)), with the corresponding ε

(γ )

ε

limiting expectations k¯ m−2 e0,i j , i, j ∈k¯ m−2 X. In what follows, we also use the following quantities, for i ∈

(γ ) k¯ m−2 e0,i

i∈

 ⎧ ⎪ j ∈ k¯ X ⎪ m−2 ⎪ ⎨ ⎪

=

(γ ) k¯ m−2 e0,i j (·)

⎪ 0 ⎪ ⎪ ⎪ ⎩

k¯ m−2 X,

for γ  ∈ [0, ∞), i = k m−1, and γ  ∈ (0, ∞], i = k m for γ  = ∞, i = k m−1, and γ  = 0, i = k m .

(12.119)

Let us define sets, for A ∈ Γ (Γ is the class of sets appearing in condition R), for k¯ m−2 X,  U[γ , k¯ m−2 q ·,u  (· , A)] · ,i  (·  ⎧ {s ≥ 0 : s(1 + γ ) ∈ U[k¯ m−2 q ·,i ⎪ k¯ m−2 u · ,i, A)]} ⎪ ⎪ ⎪ for γ  ∈ [0, ∞), i = k m−1, ⎪ ⎪ ⎪ ⎪ ⎪ [0, ∞) ⎪ ⎪ ⎨ ⎪ for γ  = ∞, i = k m−1, = ⎪ [0, ∞) ⎪ ⎪ ⎪ for γ  = 0, i = k m, ⎪ ⎪ ⎪  (·  ⎪ ⎪ {s ≥ 0 : s(1 + γ −1 ) ∈ U[k¯ m−2 q ·,i k¯ m−2 u · ,i, A)]} ⎪ ⎪  ⎪ for γ ∈ (0, ∞], i = k m . ⎩

(12.120)

  ∈ [0, ∞], A ∈ Γ, i ∈ Obviously, m(U¯ [γ , k¯ m−2 q0,u  (· , A)]) = 0, for γ k¯ m−2 X, · ,i   ¯ since m(U[k¯ m−2 q ·,i (· k¯ m−2 u · ,i, A)]) = 0, for A ∈ Γ, i ∈ k¯ m−2 X, by the condition k¯ m−2 R. The conditions k¯ m−2 R and Xγ  imply that the following relation takes place, for   s ∈ U[γ , k¯ m−2 q ·,u ,i (· , A)], γ ∈ [0, ∞], A ∈ Γ, i ∈ k¯ m−2 X, ·

 k¯ m−2 qε,uε ,i (·,

A) =

  k¯ m−2 qε,i (· uε ,

=

  k¯ m−2 qε,i (· uε,i

A) uε us  , A) −→ uε,i

(γ ) k¯ m−2 q0,i (·,

A) as ε → 0,

(12.121)

12.1 Embedded ARP

313

where, for s ≥ 0, A ∈ BZ, i ∈ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪

k¯ m−2 X,

 k¯ m−2 q0,i (s(1

+ γ ), A) 0(s) = 0 (γ )  (s(1 + γ −1 ), A) k¯ m−2 q0,i (s, A) = ⎪ q ¯ k m−2 0,i ⎪ ⎪ ⎪ 0(s) = 0 ⎩ The relation (12.122) and the condition (γ) k¯ m−2 q0,i (s, Z), s

for γ  for γ  for γ  for γ 

∈ [0, ∞), i = k m−1, = ∞, i = k m−1, ∈ (0, ∞], i = k m, = 0, i = k m

k¯ m−2 R

(12.122)

(a) imply that the function (γ)

≥ 0 is consistent with the tail probability function 1− k¯ m−2 F0,i (s), s ≥ 0 for γ ∈ [0, ∞), i = k m−1 and γ ∈ (0, ∞], i = k m . This consistency relation also holds for γ = ∞, i = k m−1 and γ = 0, i = k m , since function 0(s) = 0, s ≥ 0 is consistent with the tail probability function 1 − I(s ≥ 0), s ≥ 0. Also, the relation (12.122) and the condition k¯ m−2 R (d) imply that the function

(γ ) k¯ m−2 q0,i (·, A) is continuous almost everywhere with respect to the Lebesgue measure m(·), for B+ , for γ  ∈ [0, ∞], A ∈ Γ, i ∈ k¯ m−2 X. Therefore, the conditions k¯ m−2 I1 , k¯ m−2 J1 (k¯ m−2 J1◦ ), k¯ m−2 K1 , k¯ m−2 R, and k¯ m−2 Xγ  imply that the condition R2 is satisfied for the alternating regenerative processes (γ ) (1) (1) (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)), with the corresponding limiting functions k¯ m−2 q0,i (·, A), ε ε for γ  ∈ [0, ∞], i ∈ k¯ m−2 X, A ∈ BZ .

It is also useful to note that, according to the relations (12.119) and (12.122), the corresponding limiting stationary probabilities, ∫ ∞ 1 (γ ) (γ ) k¯ m−2 π0,i (A) = k¯ m−2 q0,i (s, A)m(ds) (γ ) 0 k¯ m−2 e0,i ∫ ∞ 1   (12.123) = k¯ m−2 q0,i (s, A)m(ds) = k¯ m−2 π0,i (A),  k¯ m−2 e0,i 0

do not depend on the parameter γ  = k¯ m−2 γ  in the cases γ  = k m−1, A ∈ BZ and γ  = k¯ m−2 γ  ∈ (0, ∞], i = k m, A ∈ BZ . The following lemma summarises the above remarks.

k¯ m−2 γ



∈ [0, ∞), i =

Lemma 12.7 Let the conditions G1 , H1 , I1 , IH,1 , J1 (J◦1 ), K1 , R, and k¯ m−2 M be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)) and, thus, con , k¯ m−2 J1 (k¯ m−2 J1◦ ), k¯ m−2 K1 , k¯ m−2 R, k¯ m−2 Xγ  ditions k¯ m−2 G1 , k¯ m−2 H1 , k¯ m−2 I1 , k¯ m−2 IH,1 (for some γ  ∈ [0, ∞]) are satisfied for the shifted alternating regenerative processes (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)). Then, the following conditions are satisfied for the com-

(1) (1) (t), k¯ m−2 ηε,u (t)), with pressed in time alternating regenerative processes (k¯ m−2 ξε,u ε ε   the time compression factor u · = k¯ m−2 u · given by relation (12.30): (i) If the condition k¯ m−2 T4 is satisfied and k¯ m−2 γ ∈ (0, ∞), then the conditions O2 , P2 , Q2 , and R2 are satisfied. (ii) If the condition k¯ m−2 T4 is satisfied and k¯ m−2 γ = 0, then the conditions O2 , P2 , Q2 , and R2 are satisfied.

314

12 Ergodic theorems for perturbed MARP

(iii) If the condition k¯ m−2 T4 is satisfied and k¯ m−2 γ = ∞, then conditions O2 , P2, Q2 , and R2 are satisfied.  (iv) If the condition k¯ m−2 T¯ 4 is satisfied and k¯ m−2 γ ∈ (0, ∞), then the conditions O2 , P¯ 2 , Q2 , and R2 are satisfied.  (v) If the condition k¯ m−2 T¯ 4 is satisfied and k¯ m−2 γ = 0, then the conditions O2 , P¯ 2 , Q2 , and R2 are satisfied. ¯  is satisfied and k¯ γ = ∞, then the conditions O2 , (vi) If the condition k¯ m−2 T m−2 4 P¯ 2, Q2 , and R2 are satisfied. (vii) The asymptotic relations (12.108), (12.109), (12.112), (12.116), and (12.121) play the role of asymptotic relations appearing in the above conditions. The corresponding quantities and sets appearing in these conditions and related relations are given by relations (12.107)–(12.123). Remark 12.3 Lemma 12.7 is, in fact, a variant of Lemmas 5.1 and 8.1, applied to the alternating regenerative processes (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)) and the compressed in

(1) time alternating regenerative processes (k¯ m−2 ξε,u  (t),

(1) k¯ m−2 ηε,uε (t)).

It follows from ˆ 2, R ˆ 2 are the above remarks, according to which the conditions O2 , Pˆ 2 or Pˆ¯ 2 , Q satisfied for the alternating regenerative processes (k¯ m−2 ξε(1) (t), k¯ m−2 ηε(1) (t)). ε

Remark 12.4 Lemma 5.1 and propositions (i)–(iii) and (vii) of Lemma 12.7 make it (1) (1) possible to apply to the alternating regenerative processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)) ε ε      ◦   Theorems 4.2–4.4, 5.1, 2.1 if conditions G1 , H1 , I1 , IH,1 , J1 (J1 ), K1 , R , k¯ m−2 M, and k¯ m−2 T4 are satisfied. In this case, the conditions k¯ m−2 Xγ  and k¯ m−2 Sβ are also satisfied. Theorem 4.2 can be applied, if γ  ∈ (0, ∞) and β  ∈ (0, ∞), Theorem 4.3 can be applied, if γ  ∈ (0, ∞) and β  = 0, Theorem 4.4 can be applied, if γ  ∈ (0, ∞) and β  = ∞, Theorem 5.1 can be applied, if γ  = 0 and β  ∈ [0, ∞), and Theorem 5.2 can be applied, if γ  = ∞ and β  ∈ (0, ∞]. The cases, where the vector parameter (γ , β ) takes value (0, ∞) or (∞, 0), are excluded, since Theorems 5.1 and 5.2 do not work in these cases for the reasons stated in Remark 5.5. Remark 12.5 Lemma 8.1 and propositions (iv)–(vii) of Lemma 12.7 make it pos(1) (1) sible to apply to the alternating regenerative processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)) ε ε  , J1 (J1◦ ), K1 , R, Theorems 6.1, 6.2, 7.1–7.6, 8.1–8.4, if conditions G1 , H1 , I1 , IH,1  ¯ k¯ m−2 M, and k¯ m−2 T4 are satisfied. In this case, the conditions k¯ m−2 Xγ and k¯ m−2 Sβ are also satisfied. Theorems 6.1 and 6.2 can be applied, if γ  ∈ (0, ∞) and β  ∈ [0, ∞], Theorem 7.1 can be applied, if γ  ∈ (0, ∞) and β  ∈ (0, ∞), Theorems 7.2 and 7.4 can be applied, if γ  ∈ (0, ∞) and β  = 0, Theorems 7.3 and 7.5 can be applied, if γ  ∈ (0, ∞) and β  = ∞, Theorem 7.6 can be applied, if γ  ∈ (0, ∞) and β  = 0 or β  = ∞, Theorem 8.1 can be applied, if γ  = 0 and β  ∈ [0, ∞), Theorem 8.2 can be applied, if γ  = ∞ and β  ∈ (0, ∞], Theorem 8.3can be applied, if γ  = 0 and β  = 0, Theorem 8.4, if γ  = ∞ and β  = ∞. The cases, where the vector parameter (γ , β ) takes value (0, ∞) or (∞, 0), are excluded, since Theorems 8.1 and 8.2 do not work in these cases for the reasons stated in Remarks 8.2 and 8.3.

12.1 Embedded ARP

315

12.1.2.5 Perturbation Conditions for Characteristics Related to Transition Periods of Compressed in Time Embedded Alternating Regenerative Processes with Partially Removed Virtual Transitions. Let us also consider the conditions associated with the compressed transition period [0, k¯ m−2 τε,uε ). The conditions IH,1 , LH,1 and the relations (11.20) imply that, for i = k m−1, k m ,  k¯ m−2 uε,k m−2  k¯ m−2 uε,i

=

 k¯ m−2 u˜ ε,k m−2  k¯ m−2 u˜ ε,i

→

 k¯ m−2 w0,k m−2,i

∈ [0, ∞] as ε → 0.

(12.124)

 , i = k m−1, k m can be calculated using Lemmas 8.21 –8.91 . The limits k¯ m−2 w0,k m−2,i Also, according to the condition k¯ m−2 M and Lemma 11.1, the following relation holds, for i ∈ k¯ m−3 K[k m−2 ] ∩ k¯ m−2 X,  k¯ m−2 w0,k m−2,i

=

k¯ m−3 w0,k m−2,i

∈ [0, ∞).

(12.125)

The conditions IH,1 , LH,1 and relations (12.90), (12.93), (12.124), and (12.125) imply that the following limits exist, for 1 ≤ n ≤ m − 2, uˇε,kn  ¯k m−2 uε

→

 k n k¯ m−2 w

∈ [0, ∞) as ε → 0,

(12.126)

where

 k n k¯ m−2 w

=

−1 −1 ⎧ ˇ 0,kn,km−2 · (k¯ m−2 w0,k + k¯ m−2 w0,k )−1 ⎪ k¯ m−2 w m−2,k m−1 m−2,k m ⎪ ⎪ ⎨ ⎪ if k¯ w  · k¯ w  > 0, m−2

⎪ 0 ⎪ ⎪ ⎪ ⎩

0,k m−2,k m−1

m−2

 if k¯ m−2 w0,k · m−2,k m−1

0,k m−2,k m

 k¯ m−2 w0,k m−2,k m

(12.127)

= 0.

The limits k¯ m−2 w k n , 1 ≤ n ≤ m−2 can also be calculated using Lemmas 8.21 –8.91 . The relation (12.86) and the conditions k¯ m−2 P2 and k¯ m−2 Xγ  imply that the following relations hold, for n = 1, . . . , m, j = k m−1, k m , k¯ m−2 pε,k n j

→

k¯ m−2 p0,k n j

as ε → 0,

(12.128)

and   k¯ m−2 Q ε,uε ,k n j (·)

= →

where  k¯ m−2 Q 0,k n j (·)

 k¯ m−2 Q ε,k n j (· uˇ ε,k n  k¯ m−2 Q 0,k n j (·)

uε uˇε,kn

)

as ε → 0,

(12.129)

316

12 Ergodic theorems for perturbed MARP

⎧ for k n ∈ Dm−1, j ∈ Dm−1, ⎪ ⎨ I(· ≥ 0)I(k n = j) ⎪ ¯ m−1, j ∈ Dm−1, w  = 0, (12.130) for k n ∈ D = I(· ≥ 0) k¯ m−2 p0,kn j kn ⎪  ⎪ ¯ F0,k j (· w −1 ) ¯ p0,k j for k n ∈ D ¯  , j ∈ D , w m−1 m−1 n n ⎩ km−2 k n k m−2 k n > 0. These relations imply that condition P 2 formulated in Sect. 10.1.2.5 also holds. The above remarks can be summarised in the following lemma. Lemma 12.8 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 J1 (J◦1 ), K1 , and k¯ m−2 M,  k¯ m−2 Xγ be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Then: (i) The condition P 2 is satisfied for the compressed in time multi-alternating regenerative processes (ξε,uε (t), ηε,uε (t)), with transition periods [0, k¯ m−2 τε,uε ) and the time compression factor u · = k¯ m−2 u · given by relation (12.94). (ii) The asymptotic relations (12.128) and (12.129) play the role of asymptotic relation appearing in the condition P 2 . The corresponding quantities and sets appearing in this condition are given by the relations (12.126), (12.127), and (12.130).

12.2 Embedded Regenerative Processes In this section, we describe embedded regenerative processes built using recurrent algorithms of time–space aggregation for regeneration times based on total and partial removing of virtual transition and reduction of phase space for modulating semi-Markov processes.

12.2.1 Embedded Regenerative Processes Based on Total Removing of Virtual Transitions 12.2.1.1 Embedded Regenerative Processes with Transition Period and Totally Removed Virtual Transitions. Let (ξε (t), ηε (t)) be, for every ε ∈ (0, 1], the multialternating regenerative process with the phase space Z × X defined in Sect. 10.1.1. We assume that the conditions G1 , H1 I1 , IH,1 , L1 , LH,1 , J1 , K1 , and R hold for these processes. Also, let (ηε,n, κε,n ) be the corresponding Markov renewal process used to construct the modulating semi-Markov process ηε (t). According to the remarks made in Sect. 11.2.1.3, there exists a sequence of states k¯ n = k1, . . . , k m−2 in the phase space X = {1, . . . , m}, such that the condition k¯ m−2 M is satisfied. This case was considered in Sects. 12.1.1 and 12.1.2. However, according to the remarks made in Sect. 11.2.1.3, there also exists a sequence of states k¯ n = k1, . . . , k m−1 in the phase space X = {1, . . . , m}, such that the condition k¯ m−1 M is satisfied.

12.2 Embedded regenerative processes

317

The condition k¯ m−1,1 M means that the condition k¯ m−2 M holds, and in addition, the following relation holds: k¯ m−2 u˜ ε,k m−1 k¯ m−2 u˜ ε,i

→

k¯ m−2 w0,k m−1,i

∈ [0, ∞) as ε → 0, for i = k m−1, k m,

(12.131)

where, for i = k m−1, k m , k¯ m−2 u˜ ε,i

=

m−2 

−1 k¯r p¯ε,ii uε,i .

(12.132)

r=0

The condition k¯ m−1 M means that state k m is the most absorbing state in the phase space X. Let us also assume the random variable ηε = ηε (0) satisfy condition k¯ m−1 W, i.e., P{ηε (0) = k m } = 1, for ε ∈ (0, 1]. Let (k¯ m−1 ξε (t), k¯ m−1 ηε (t)) be the multi-alternating regenerative processes with the reduced phase space k¯ m−1 X = {k m }of the modulating semi-Markov process k¯ m−1 ηε (t) built using the recurrent algorithm for time–space aggregation of regeneration times (described in Sect. 11.2) applied to the multi-alternating regenerative processes (ξε (t), ηε (t)). In this case, the modulating semi-Markov process k¯ m−1 ηε (t) has the one-state phase n space k¯ m−1 X = {k m } and moments of jumps k¯ m−1 ζε,n = l=1 k¯ m−1 κε,n, n = 0, 1, . . ., which are sums of i.i.d. random variables k¯ m−1 κε,n, n = 1, 2, . . .. This process makes only virtual transitions of the form k m → k m at moments k¯ m−1 ζε,n , and, thus, P{k¯ m−1 ηε (t) = k m, t ≥ 0} = 1. It should be noted that, according to the recurrent algorithm described in Sect. 11.2, virtual transitions are totally removed for all intermediate reduced modulating semi-Markov processes but not for the resulting semi-Markov process k¯ m−1 ηε (t). In this case, the process (k¯ m−1 ξε (t), k¯ m−1 ηε (t)), t ≥ 0, built using the recurrent algorithm for time–space aggregation of regeneration times (described in Sect. 11.2) applied to the multi-alternating regenerative processes (ξε (t), ηε (t)), is a regenerative process with regeneration times k¯ m−1 ζε,n . Since the modulating component k¯ m−1 ηε (t) is a degenerate process, the first component k¯ m−1 ξε (t) = ξε (t) is itself a regenerative process with regeneration times k¯ m−1 ζε,n . Let us now consider the general case when the condition k¯ m−1 W is not satisfied. We define the following stopping moments, for ε ∈ (0, 1],  0 if ηε,0 ∈ k¯ m−1 X, (12.133) k¯ m−1 αε = min(n ≥ 1 : η ∈ X) if ηε,0  k¯ m−1 X, ε,n k¯ m−1 and k¯ m−1 τε

=

k¯ m−1 αε

n=1

κε,n .

(12.134)

318

12 Ergodic theorems for perturbed MARP

The process (ξε (t), ηε (t)) can be considered as a multi-alternating regenerative process with a transition period [0, k¯ m−1 τε ). Consider the shifted multi-alternating regenerative process, (ξε(1) (t), ηε(1) (t)) = (ξε (k¯ m−1 τε + t), ηε (k¯ m−1 τε + t)).

(12.135)

Obviously, the condition k¯ m−1 W is satisfied for these multi-alternating regenerative processes. Let (k¯ m−1 ξε(1) (t), k¯ m−1 ηε(1) (t)) be the regenerative process (with the one-state phase

space k¯ m−1 X = {k m } for the modulating semi-Markov processes k¯ m−1 ηε(1) (t))) built using the recurrent algorithm for time–space aggregation of regeneration times (described in Sect. 11.2) applied to the multi-alternating regenerative process (ξε(1) (t), ηε(1) (t)). According to Theorem 11.1, the following relation takes place, for i ∈ k¯ m−1 X, t ≥ 0, A ∈ BZ and ε ∈ (0, 1], Pε(1) (t, A) = P{ξε(1) (t) ∈ A} = P{k¯ m−1 ξε(1) (t) ∈ A}.

(12.136)

The probability Pε(1) (t, A) = P{ξε(1) (t) ∈ A} is, for every A ∈ BZ , a measurable function of t ≥ 0 and the only bounded solution for the following renewal equation, ∫ t (1) Pε (t, A) = k¯ m−1 qε,km (t, A) + Pε(1) (t − s, A) k¯ m−1 Fε,km (ds), t ≥ 0, (12.137) 0

where, for A ∈ BZ, t ≥ 0, k¯ m−1 qε,k m (t,

A) = Pkm {k¯ m−1 ξε (t) ∈ A,

k¯ m−1 ζε,1

> t},

(12.138)

and k¯ m−1 Fε,k m (t)

= Pkm { k¯ m−1 ζε,1 ≤ t}.

(12.139)

Also, the probabilities Pε,i (t, A) = P{ξε (t) ∈ A}, i ∈ X are, for every A ∈ BZ , measurable functions of t ≥ 0, which satisfy the following renewal type relations: Pε,i (t, A) = +

k¯ m−1 qε,i (t, A) t Pε(1) (t − s, 0

∫

A) k¯ m−1 Fε,i (ds), t ≥ 0, i ∈ X,

(12.140)

where the functions k¯ m−2 qε,i (t, A) and the distribution functions k¯ m−1 Fε,i (t) are given, for A ∈ BZ, t ≥ 0, i ∈ X, by the following relations: k¯ m−1 qε,i (t,

and

A) = Pi {ξε (t) ∈ A,

 k¯ m−1 Fε,i (t)

k¯ m−1 τε

= Pi {k¯ m−1 τε ≤ t}.

> t}

(12.141) (12.142)

12.2 Embedded regenerative processes

319

12.2.1.2 Perturbation Conditions for Embedded Regenerative Processes with Totally Removed Virtual Transitions. The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R, and k¯ m−1 M assumed to be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)), are also satisfied for the shifted multi-alternating regenerative processes (ξε(1) (t), ηε(1) (t)). The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R take for the regenerative processes (k¯ m−1 ξε(1) (t), k¯ m−1 ηε(1) (t)) the forms described below. The fact that the space k¯ m−1 X = {k m } is a one-state set should be taken into account. The condition k¯ m−1 G1 takes the following form: k¯ m−1 G1 : k¯ m−1 Fε,k m (0)

= 0, ε ∈ (0, 1].

Here, it is taken into account that the transition probability k¯ m−1 pε,km km ≡ 1 and, thus, the condition k¯ m−1 G1 (a) is automatically satisfied. The conditions k¯ m−1 H1 , k¯ m−1 I1 , and k¯ m−1 IH,1 (here, H is the complete family of asymptotically comparable functions appearing in the condition IH,1 ) automatically hold since, in this case, the phase space k¯ m−1 X = {k m } and the transition probability k¯ m−1 pε,k m k m ≡ 1. We denote, for ε ∈ (0, 1], ∫ ∞ e−su k¯ m−1 Fε,km (du), s ≥ 0, (12.143) k¯ m−1 φε,k m (s) = 0

∫

and k¯ m−1 fε,k m

= 0

∞

u k¯ m−1 Fε,km (du).

(12.144)

The conditions k¯ m−1 J1 , k¯ m−1 J◦1 , and k¯ m−1 K1 take the following forms: k¯ m−1 J1 :

(a) k¯ m−1 Fε,km (· k¯ m−1 uε,km ) ⇒ k¯ m−1 F0,km (·) as ε → 0, (b) k¯ m−1 F0,km (·) is a non-arithmetic distribution function without singular component, (c) k¯ m−1 uε,k m ∈ (0, ∞), ε ∈ (0, 1].

And, equivalently, ◦ k¯ m−1 J1 :

k¯ m−1 φε,k m (s/ k¯ m−1 uε,k m ) → k¯ m−1 φ0,k m (s) as ε → 0, for s ≥ 0, (b) ¯k m−1 φ0,k m (·) is the Laplace transform of a non-arithmetic distribution function without singular component k¯ m−1 F0,km (·), (c) k¯ m−1 uε,km ∈ (0, ∞), ε ∈ (0, 1].

k¯ m−1 K1 :

(a) k¯ m−1 fε,km < ∞, for ε ∈ (0, 1], (b) k¯ m−1 fε,km / k¯ m−1 uε,km → k¯ m−1 f0,km < ∞ as ε → 0, (c) k¯ m−1 f0,km is the first moment of the distribution function k¯ m−1 F0,km (·).

(a)

The conditions k¯ m−1 L1 and k¯ m−2 LH,1 take the following forms: k¯ m−1 L1 : k¯ m−1 uε,k m

and

→

k¯ m−1 u0,k m

∈ (0, ∞] as ε → 0,

320

12 Ergodic theorems for perturbed MARP

Function k¯ m−1 u ·,km belongs to the complete family of asymptotically comparable functions H appearing in condition LH,1 .

k¯ m−2 LH,1 :

Finally, the condition k¯ m−1 R takes the following form: k¯ m−1 R:

There exists a function k¯ m−1 q0,km (t, A), t ≥ 0, A ∈ BZ , which belongs to the class P[BZ ], a class of sets Γ ⊆ BZ , and Borel sets U[k¯ m−1 q ·,km (· k¯ m−1 u ·,km , A)], A ∈ Γ such that: (a) the function k¯ m−1 q0,km (t, A), t ∈ R+, A ∈ BZ is consistent with the tail probability function 1 − k¯ m−1 F0,km (t), t ∈ R+ ; (b) the us

functions k¯ m−1 qε,km (· k¯ m−1 uε,km , A) −→ k¯ m−1 q0,km (·, A) as ε → 0, for points s ∈ U[k¯ m−1 q ·,km (· k¯ m−1 u ·,km , A)], A ∈ Γ; (c) m(U¯ [k¯ m−1 q ·,km (· k¯ m−1 u ·,km , A)]) = 0, for A ∈ Γ; (d) the function k¯ m−1 q˜0,km (·, A) is continuous almost everywhere with respect to the Lebesgue measure m(·) on B+ , for A ∈ Γ. The normalisation function and takes the following form:

k¯ m−1 uε,k m

is given by the relations (11.10)–(11.12)

=

k¯ m−2 u˜ ε,k m

=

k¯ m−1 uε,k m

m−2 

−1 k¯r p¯ε,k m k m uε,k m .

(12.145)

r=0

The following lemma, which is a direct corollary of Theorem 11.1, takes place. Lemma 12.9 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R, and k¯ m−1 M be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Then: (i) The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R are satisfied for the shifted regenerative processes (k¯ m−1 ξε(1) (t), k¯ m−1 ηε(1) (t)) in the form of conditions ◦ k¯ m−1 G1 , k¯ m−1 H1 , k¯ m−1 I1 , k¯ m−1 IH,1 , k¯ m−1 L1 , k¯ m−1 LH,1 , k¯ m−1 J1 (k¯ m−1 J1 ), k¯ m−1 K1 , k¯ m−1 R. (ii) The quantities and sets appearing in the above conditions can be calculated using the recurrent algorithms described in Lemmas 10.4–10.10, 10.19–10.24, and Theorem 11.1. (iii) The normalisation function k¯ m−1 u ·,km is given by the relation (12.145). 12.2.1.3 Perturbation Conditions for Characteristics Related to Transition Periods of Embedded Regenerative Processes with Totally Removed Virtual Transitions. Let us also analyse the conditions associated with the transition period [0, k¯ m−1 τε ). Let us use the notation, Dm =

k¯ m−1 X

¯ m = {k1, . . . , k m−1 }. = {k m }, D

(12.146)

The stopping time k¯ m−1 τε is connected with the first hitting time τε,Dm in the domain Dm (for the semi-Markov process ηε (t)) by the following relation: k¯ m−1 τε

= I(ηε (0) ∈

¯ k¯ m−1 X) τε,D m .

(12.147)

12.2 Embedded regenerative processes

321

Let us introduce the distribution functions, for k n ∈ X,  k¯ m−1 Fε,k n (·)

= Pkn {k¯ m−1 τε ≤ ·}  =

I(· ≥ 0) for k n ∈ Dm, ¯ m. Pk n {τε,D m ≤ ·} for k n ∈ D

(12.148)

In this case, for k n ∈ X and ε ∈ (0, 1], the hitting probability, k¯ m−1 pε,k n,k m

= Pkn {ηε (k¯ m−1 τε ) = k m } = 1.

(12.149)

From Theorems 10.21 –10.41 and the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , ¯ m, K1 , and k¯ m−1 M it follows that the following relation holds, for k n ∈ D Pk n {τε,D m ≤ · uˇ ε,k n } ⇒ F0,D m,k n k m (·) as ε → 0,

(12.150)

¯ m are given by the relation where: (a) the normalisation functions uˇ ·,kn , k n ∈ D ¯ (10.115)1 ; (b) F0,Dm,kn km (·), k n ∈ Dm are the distribution functions concentrated on the interval [0, ∞), with Laplace transforms given by the relations (10.125)1 and (10.131)1 . ¯ m−1, j ∈ Dm−1 , It is natural to denote, for k n ∈ D  I(· ≥ 0) for k n ∈ Dm, j ∈ Dm,  (12.151) F (·) = k¯ m−1 0,k n ¯ m, j ∈ Dm . F0,Dm,kn km (·) for k n ∈ D The relation (12.20) implies that  k¯ m−1 Fε,k n (· uˇ ε,k n )

= Pkn {k¯ m−1 τε ≤ · uˇε,kn } ⇒ k¯ F0,kn (·) as ε → 0. m−1

(12.152)

The relation (12.24) implies that the following condition related to the transition period [0, k¯ m−2 τε ) is satisfied:  k¯ m−1 P2 : k¯ m−1 Fε,k n (· uˇ ε,k n )

⇒

 k¯ m−1 F0,k n (·)

as ε → 0, for n = 1, . . . , m.

It is important that, according to the relation (10.88)1 , uˇε,km−1 =

m−2 

−1 k¯l p¯ε,k m−1 k m−1 uε,k m−1 ,

(12.153)

l=0

and, according to the relation (10.118)1 given in Lemma 10.151 , for any 1 ≤ n ≤ m − 1, uˇε,kn → k¯ m−1 wˇ 0,kn,km−1 ∈ [0, ∞) as ε → 0. (12.154) uˇε,km−1 The relations (11.12), (12.132), and (12.153) imply that uˇε,km−1 =

k¯ m−2 u˜ ε,k m−1

and k¯ m−1 uε,km =

k¯ m−2 u˜ ε,k m .

(12.155)

322

12 Ergodic theorems for perturbed MARP

Thus, the condition k¯ m−1 M implies that uˇε,km−1 → k¯ m−1 uε,k m

k¯ m−2 w0,k m−1,k m

∈ [0, ∞) as ε → 0.

(12.156)

Lemma 12.10 Let the conditions G1 , H1 , I1 , IH,1 , J1 (J◦1 ), K1 , and k¯ m−1 M be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)) with the transition periods [0, k¯ m−1 τε ). Then: (i) The condition k¯ m−1 P2 is satisfied. (ii) The quantities and sets appearing in the condition k¯ m−1 P2 can be calculated using the recurrent algorithms described in Lemmas 10.131 –10.141 and Theorems 10.21 –10.41 . (iii) The normalisation functions uˇ ·,kn , n = 1, . . . , m − 1, are given by the relation (10.115)1 . (iv) The asymptotic relations (12.154) and (12.156) hold. 12.2.1.4 Compressed in Time Embedded Regenerative Processes with Totally Removed Virtual Transitions. Due to the conditions IH,1 and LH,1 , the normalisation functions uˇ ·,kn , 1 ≤ n ≤ m−1, and k¯ m−1 u ·,km belong to the class of asymptotically comparable functions H. This allows the use, in this case, of the universal normalisation functions, uˆε =

k¯ m−1 uε,k m

=

k¯ m−2 u˜ ε,k m .

(12.157)

Let us consider the multi-alternating regenerative process compressed in time, (ξε, uˆ ε (t), ηε, uˆ ε (t)) = (ξε (t uˆε ), ηε (t uˆε )), t ≥ 0.

(12.158)

The process (ξε, uˆ ε (t), ηε, uˆ ε (t)) can be considered as a multi-alternating regenerative process with a transition period [0, k¯ m−1 τε, uˆ ε ), where k¯ m−1 τε, uˆ ε

=

k¯ m−1 τε

uˆε

.

(12.159)

Let us also consider the corresponding shifted multi-alternating regenerative process compressed in time, (1) (1) (ξε, uˆ ε (t), ηε, uˆ ε (t)) = (ξε, uˆ ε (k¯ m−1 τε, uˆ ε + t), ηε, uˆ ε (k¯ m−1 τε, uˆ ε + t)), t ≥ 0.

(12.160)

The condition k¯ m−1 W is satisfied for these multi-alternating regenerative processes. (1) (1) Let, also, (k¯ m−1 ξε, uˆ ε (t), k¯ m−1 ηε, uˆ ε (t)) be the regenerative process with removed virtual transitions (with the reduced phase space k¯ m−1 X = {k m } of the modulating

(1) semi-Markov process k¯ m−1 ηε, uˆ ε (t)) built using the recurrent algorithm for time– space aggregation of regeneration times (described in Sect. 11.2) applied to the (1) (1) multi-alternating regenerative process (ξε, uˆ ε (t), ηε, uˆ ε (t)).

12.2 Embedded regenerative processes

323

The following relation, which is equivalent to the relation (12.136), takes place, for t ≥ 0, A ∈ BZ and ε ∈ (0, 1], (1) (1) (1) Pε, uˆ ε (t, A) = Pi {ξε, uˆ ε (t) ∈ A} = P {k¯ m−1 ξε, uˆ ε (t) ∈ A}

= Pε(1) (t uˆε, A) = P{ξε(1) (t uˆε ) ∈ A} = P{k¯ m−1 ξε(1) (t uˆε ) ∈ A}. (12.161) The relations (12.137)–(12.142) for the multi-alternating regenerative process (ξε (t), ηε (t)) and the regenerative process (k¯ m−1 ξε(1) (t), k¯ m−1 ηε(1) (t)) can be re-written for this multi-alternating regenerative process compressed in time with factor uˆε . (1) The probability Pε, uˆ ε (t, A) is, for every A ∈ BZ , a measurable function of t ≥ 0 and the only bounded solution for the following renewal equation (which is an equivalent variant of the renewal equation (12.137)), (1) Pε, uˆ ε (t, A) = k¯ m−1 qε, uˆ ε (t, A) ∫ t (1) + Pε, uˆ ε (t − s, A) k¯ m−1 Fε, uˆ ε ,k m (ds), t ≥ 0,

(12.162)

0

where, for A ∈ BZ, t ≥ 0, k¯ m−1 qε, uˆ ε ,i (t,

A) = Pkm {ξε (t uˆε ) ∈ A,

k¯ m−1 ζε,1

> t uˆε },

(12.163)

and k¯ m−1 Fε, uˆ ε ,k m (t)

= Pkm { k¯ m−1 ζε,1 ≤ t uˆε }.

(12.164)

Also, the probabilities Pε, uˆ ε ,i (t, A) = Pi {ξε, uˆ ε (t) ∈ A}, i ∈ X are, for every A ∈ BZ , measurable functions of t ≥ 0, which satisfy the following renewal type relations (which are equivalent variant of the renewal type relations (12.140)), Pε, uˆ ε ,i (t, A) = k¯ m−1 qε, uˆ ε ,i (t, A) ∫ t (1)  Pε, + uˆ ε (t − s, A) k¯ m−1 Fε, uˆ ε ,i (ds), t ≥ 0, i ∈ X,

(12.165)

0

where, for A ∈ BZ, t ≥ 0, i ∈ X, k¯ m−2 qε, uˆ ε ,i (t,

and

A) = Pi {ξε (t uˆε ) ∈ A,

 k¯ m−1 Fε, uˆ ε ,i (t)

k¯ m−2 τε

> t uˆε },

= Pi {k¯ m−1 τε ≤ t uˆε }.

(12.166) (12.167)

The transition characteristics for the compressed in time regenerative process (1) (1) (k¯ m−1 ξε, uˆ ε (t), k¯ m−1 ηε, uˆ ε (t)) are connected with the transition characteristics that we

have for the regenerative process (k¯ m−1 ξε(1) (t), for ε ∈ (0, 1],

(1) k¯ m−1 ηε (t))

by the following relations,

324

12 Ergodic theorems for perturbed MARP k¯ m−1 Q ε, uˆ ε ,k m k m (t) k¯ m−1 pε, uˆ ε ,k m k m

=

k¯ m−1 ψε, uˆ ε ,k m k m (s)

=

=

k¯ m−1 Fε, uˆ ε ,k m (t)

k¯ m−1 pε,k m k m

=

k¯ m−1 eε, uˆ ε ,k m k m

= uˆ−1 ε k¯ m−1

=

k¯ m−1 Fε,k m (t uˆ ε ),

t ≥ 0,

= 1,

k¯ m−1 φε, uˆ ε ,k m (s) ∫ ∞ −su/uˆ ε

k¯ m−1 φε,k m (s/uˆ ε )

=

e

0

k¯ m−1 Fε,k m (du), s

≥ 0,

=

k¯ m−1 fε, uˆ ε ,k m ∫ ∞ fε,km = uˆ−1 u k¯ m−1 Fε,km (du), ε 0

k¯ m−1 qε, uˆ ε ,k m (t,

A) =

k¯ m−1 qε,k m (t uˆ ε ,

A), t ≥ 0, A ∈ BZ .

(12.168)

Let us prove that the perturbation conditions formulated in Sects. 1.1.1.1 and 2.1.2.1 (1) (1) are satisfied for the shifted regenerative processes (k¯ m−1 ξε, uˆ ε (t), k¯ m−1 ηε, uˆ ε (t)). Note that the conditions IH,1 and LH,1 imply that uˆε → uˆ0 ∈ (0, ∞] as ε → 0.

(12.169)

The condition k¯ m−1 G1 implies that the condition O1 is satisfied. Since k¯ m−1 pε, uˆ ε ,km km ≡ 1, the condition P1 (a) is automatically satisfied, with the corresponding limiting probability k¯ m−1 p0, uˆ 0,km km = 1. The conditions k¯ m−1 I1 and k¯ m−1 J1 (k¯ m−1 J◦1 ) imply that the following relation takes place: k¯ m−1 Fε, uˆ ε ,k m (·)

=

k¯ m−1 Q ε,k m k m (· uˆ ε )

⇒

k¯ m−1 F0,k m (·)

=

k¯ m−1 Q 0,k m k m (·)

as ε → 0.

(12.170)

The relation (12.170) is equivalent to the following convergence relation for the Laplace transforms k¯ m−1 ψε,km,km (·/uˆε ) of the distribution functions Q ε,km km (· uˆε ), k¯ m−1 φε, uˆ ε ,k m (s)

→

=

k¯ m−1 ψε,k m k m (s/uˆ ε )

k¯ m−1 φ0,k m (s)

=

k¯ m−1 ψ0,k m k m (s)

as ε → 0, for s ≥ 0.

(12.171)

Note that the condition k¯ m−1 J1 (b) implies that the distribution function is weakly non-arithmetic. The above remarks imply that the condition P1 is satisfied for the regenerative (1) (1) processes (k¯ m−1 ξε, uˆ ε (t), k¯ m−1 ηε, uˆ ε (t)). The conditions k¯ m−1 I1 , k¯ m−1 J1 (k¯ m−1 J◦1 ), and k¯ m−1 K1 imply that the following relation takes place: k¯ m−1 F0,u0,k m (·)

k¯ m−1 fε, uˆ ε ,k m

= →

k¯ m−1 eε,k m k m /uˆ ε . k¯ m−1 f0,k m

=

k¯ m−1 e0,k m k m

as ε → 0.

(12.172)

12.2 Embedded regenerative processes

325

Taking into account the above remarks, Lemma 10.2 and the relation (12.172), we can conclude that the conditions k¯ m−1 I1 , k¯ m−1 J1 (k¯ m−1 J◦1 ), and k¯ m−1 K1 imply that the (1) (1) condition Q1 is satisfied for the regenerative processes (k¯ m−1 ξε, uˆ ε (t), k¯ m−1 ηε, uˆ ε (t)). We denote

U[k¯ m−1 q ·, uˆ ·,k m (· , A)] = U[k¯ m−1 q ·,k m (· k¯ m−1 uˆ · ,k m , A)].

The condition

k¯ m−1 R

U[k¯ m−1 q ·, uˆ ·,k m (· , A)],

(12.173)

implies that the following relation takes place, for s ∈

k¯ m−1 qε, uˆ ε ,k m (·,

A) = us

k¯ m−1 qε,k m (· k¯ m−1 uˆ ε,k m ,

−→

k¯ m−1 q0,k m (·,

A)

A) as ε → 0.

(12.174)

The relation (12.174) and the condition k¯ m−1 R (a) imply that the function is consistent with the tail probability function 1 − k¯ m−1 F0,km (s). Also, the relation (12.174) and the condition k¯ m−1 R (d) imply that the function k¯ m−1 q0,k m (·, A) is continuous almost everywhere with respect to the Lebesgue measure m(·), for B+ , for A ∈ Γ. Therefore, the conditions k¯ m−1 I1 , k¯ m−1 J1 (k¯ m−1 J◦1 ), k¯ m−1 K1 , and k¯ m−1 R imply that the k¯ m−1 q0,k m (s, Z)

(1) (1) condition R1 is satisfied for the regenerative processes (k¯ m−1 ξε, uˆ ε (t), k¯ m−1 ηε, uˆ ε (t)). The following lemma summarises the above remarks.

Lemma 12.11 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R, and k¯ m−1 M be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)) and, thus, conditions k¯ m−1 G1 , k¯ m−1 H1 , k¯ m−1 I1 , k¯ m−1 IH,1 , k¯ m−1 L1 , k¯ m−1 LH,1 , k¯ m−1 J1 (k¯ m−1 J◦1 ), k¯ m−1 K1 , and k¯ m−1 R are satisfied for the shifted regenerative processes

(k¯ m−1 ξε(1) (t), k¯ m−1 ηε(1) (t)). Then: (i) The conditions O1 , P1 , Q1 , and R1 are satisfied for the compressed in time (1) (1) regenerative processes (k¯ m−1 ξε, uˆ ε (t), k¯ m−1 ηε, uˆ ε (t)), with the time compression factor uˆ · = k¯ m−1 uˆ · given by the relation (12.157). (ii) The asymptotic relations (12.170), (12.171), (12.172), and (12.174) play the roles of asymptotic relations appearing in the above conditions. The corresponding quantities and sets appearing in these conditions are given by the above relations and the relations (12.168) and (12.173).

12.2.1.5 Perturbation Conditions for Characteristics Related to Transition Periods of Compressed in Time Embedded Regenerative Processes with Totally Removed Virtual Transitions. Let us also consider the conditions associated with the compressed transition period [0, k¯ m−1 τε, uˆ ε ). The condition k¯ m−1 P2 and the relation (12.156) imply that the following relation holds, for n = 1, . . . , m,

326

12 Ergodic theorems for perturbed MARP

 k¯ m−1 Fε,k n (· uˆ ε )

= ⇒

 k¯ m−1 Fε,k n (· uˇ ε,k n

uˆε uˇε,kn

)

−1  k¯ m−1 F0,k n (· k¯ m−2 w0,k m−1,k m )

as ε → 0.

(12.175)

This relation implies that the condition P 1 holds for the multi-alternating regenerative processes (ξε, uˆ ε (t), ηε, uˆ ε (t)) with the transition periods [0, k¯ m−1 τε, uˆ ε ). The above remarks can be summarised in the following lemma. Lemma 12.12 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , and k¯ m−1 M be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Then: (i) The condition P 1 is satisfied for the compressed in time regenerative processes (ξε, uˆ ε (t), ηε, uˆ ε (t)) with the transition periods [0, k¯ m−1 τε, uˆ ε ) and the time compression factor uˆ · = k¯ m−1 u ·,km given by the relation (12.145). (ii) The asymptotic relation (12.175) plays the role of asymptotic relation appearing in the condition P 1 . The corresponding quantities and sets appearing in this condition are given by relations (12.151), (12.156), and (12.175). Lemmas 12.11 and 12.12 make it possible to apply Theorems 2.1–2.3 to the regenerative processes (k¯ m−1 ξε, uˆ ε (t), k¯ m−1 ηε, uˆ ε (t)), if the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R, k¯ m−1 M are satisfied.

12.2.2 Embedded Regenerative Processes Based on Partial Removing of Virtual Transitions 12.2.2.1 Embedded Regenerative Processes with Transition Period and Partially Removed Virtual Transitions. Let (ξε (t), ηε (t)) be, for every ε ∈ (0, 1], the multi-alternating regenerative process with the phase space Z × X defined in Sect. 10.1.1. We assume that the conditions G1 , H1 I1 , IH,1 , L1 , LH,1 , J1 , K1 , and R are satisfied for these processes. Also, let (ηε,n, κε,n ) be the corresponding Markov renewal process used to construct the modulating semi-Markov process ηε (t). As in Sect. 12.2.1.1, we assume that the condition k¯ m−1 M is satisfied. In this case, the state k m is the most absorbing state in the phase space X. Let us also assume the random variable ηε = ηε (0) satisfy the condition k¯ m−1 W, i.e., P{ηε (0) = k m } = 1, for ε ∈ (0, 1]. Let (k¯ m−1 ξε (t), k¯ m−1 ηε (t)) be the multi-alternating regenerative processes with the reduced phase space of modulating semi-Markov process k¯ m−1 X = {k m } built using the recurrent algorithm for time–space aggregation of regeneration times (described in Sect. 11.3) applied to the multi-alternating regenerative processes (ξε (t), ηε (t)). In this case, the modulating semi-Markov process k¯ m−1 ηε (t)has the one-state n  =  phase space k¯ m−1 X = {k m } and the moments of jumps k¯ m−1 ζε,n l=1 k¯ m−1 κε,n, n =  0, 1, . . ., which are sums of i.i.d. random variables k¯ m−1 κε,n, n = 1, 2, . . .. This process

12.2 Embedded regenerative processes

327

makes only virtual transitions of the form k m → k m at the moments k¯ m−1 ζε,n , and, thus, P{k¯ m−1 ηε (t) = k m, t ≥ 0} = 1. It should be noted that, according to the recurrent algorithm described in Sect. 11.3, the virtual transitions are partially removed for all intermediate reduced semi-Markov processes but not for the resulting semi-Markov process k¯ m−1 ηε (t). In this case, the process (k¯ m−1 ξε (t), k¯ m−1 ηε (t)), t ≥ 0, built using the recurrent algorithm for time–space aggregation of regeneration times (described in Sect. 11.3) applied to the multi-alternating regenerative processes (ξε (t), ηε (t)), is a regenerative  . Since the modulating component  process with regeneration times k¯ m−1 ζε,n k¯ m−1 ηε (t)  is a degenerate process, the first component k¯ m−1 ξε (t) = ξε (t) is itself a regenerative  . process with regeneration times k¯ m−1 ζε,n Let us now consider the general case when the condition k¯ m−1 W is not satisfied. Let us define, for ε ∈ (0, 1], the stopping moments,  0 if ηε,0 ∈ k¯ m−1 X, (12.176) k¯ m−1 αε = min(n ≥ 1 : η ∈ X) if ηε,0  k¯ m−1 X, ¯ ε,n k m−1 and k¯ m−1 τε =

k¯ m−1 αε



κε,n .

(12.177)

n=1

The process (ξε (t), ηε (t)) can be considered as the multi-alternating regenerative process with transition period [0, k¯ m−1 τε ). Let us consider the shifted multi-alternating regenerative process, (ξε(1) (t), ηε(1) (t)) = (ξε (k¯ m−1 τε + t), ηε (k¯ m−1 τε + t)).

(12.178)

Obviously, condition k¯ m−1 W is satisfied for these multi-alternating regenerative processes. Let, also, (k¯ m−1 ξε(1) (t), k¯ m−1 ηε(1) (t)) be the regenerative processes (with the onestate phase space k¯ m−1 X = {k m } for the modulating semi-Markov processes (1) k¯ m−1 ηε (t))), built using the recurrent algorithm for time–space aggregation of regen-

eration times (described in Sect. 11.3) applied to the multi-alternating regenerative processes (ξε(1) (t), ηε(1) (t)). According to Theorem 11.2, the following relation takes place, for i ∈ k¯ m−1 X, t ≥ 0, A ∈ BZ and ε ∈ (0, 1], Pε(1) (t, A) = P{ξε(1) (t) ∈ A} = P{k¯ m−1 ξε(1) (t) ∈ A}.

(12.179)

The probability Pε(1) (t, A) = P{ξε(1) (t) ∈ A} is, for every A ∈ BZ , a measurable function of t ≥ 0 and the only bounded solution for the following renewal equation: ∫ t (1)   Pε (t, A) = k¯ m−1 qε,km (t, A) + Pε(1) (t − s, A) k¯ m−1 Fε,k (ds), t ≥ 0, (12.180) m 0

328

12 Ergodic theorems for perturbed MARP

where, for A ∈ BZ, t ≥ 0,  k¯ m−1 qε,k m (t,

and

A) = Pkm {k¯ m−1 ξε (t) ∈ A,  k¯ m−1 Fε,k m (t)

 k¯ m−1 ζε,1

> t},

 = Pkm { k¯ m−1 ζε,1 ≤ t}.

(12.181) (12.182)

Also, the probabilities Pε,i (t, A) = P{ξε (t) ∈ A}, i ∈ X are, for every A ∈ BZ , measurable functions of t ≥ 0, which satisfy the following renewal type relations: Pε,i (t, A) = +

k¯ m−1 qε,i (t, A) t Pε(1) (t − s, 0

∫

A) k¯ m−1 Fε,i (ds), t ≥ 0, i ∈ X,

where functions k¯ m−2 qε,i (t, A) and distribution functions A ∈ BZ, t ≥ 0, i ∈ X, by relations, k¯ m−1 qε,i (t,

and

A) = Pi {ξε (t) ∈ A,

 k¯ m−1 Fε,i (t)

k¯ m−1 τε

 k¯ m−1 Fε,i (t) > t}

= Pi {k¯ m−1 τε ≤ t}.

(12.183) are given, for (12.184) (12.185)

12.2.2.2 Perturbation Conditions for Embedded Regenerative Processes with Partially Removed Virtual Transitions. The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R, and k¯ m−1 M assumed to be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)) are also satisfied for the shifted multi-alternating regenerative processes (ξε(1) (t), ηε(1) (t)). The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R take for the regenerative processes (k¯ m−1 ξε(1) (t), k¯ m−1 ηε(1) (t)) the forms described below. The fact that space k¯ m−1 X = {k m } is a one-state set should be taken into account. The condition k¯ m−1 G1 takes the following form:   k¯ m−1 G1 : k¯ m−1 Fε,k m (0)

= 0, ε ∈ (0, 1].

 Here, it is taken into account that transition probability k¯ m−1 pε,k ≡ 1 and, m km  thus, the condition k¯ m−1 G1 (a) is automatically satisfied.  (here, H is the complete family The conditions k¯ m−1 H1 , k¯ m−1 I1 , and k¯ m−1 IH,1 of asymptotically comparable functions appearing in condition IH,1 ) are automatically satisfied since, in this case, the phase space k¯ m−1 X = {k m } and the transition  ≡ 1. probability k¯ m−1 pε,k m km We denote, for ε ∈ (0, 1], ∫ ∞   e−su k¯ m−1 Fε,k (du), s ≥ 0, (12.186) k¯ m−1 φε,k m (s) = m 0

∫

and  k¯ m−1 fε,k m

= 0

∞

 u k¯ m−1 Fε,k (du). m

(12.187)

12.2 Embedded regenerative processes

329

The conditions k¯ m−1 J1 , k¯ m−1 J1◦ , and k¯ m−1 K1 take the following forms:  k¯ m−1 J1 :

    (a) k¯ m−1 Fε,k (· k¯ m−1 uε,k ) ⇒ k¯ m−1 F0,k (·) as ε → 0, (b) k¯ m−1 F0,k (·) is a m m m m non-arithmetic distribution function without singular component, (c)  k¯ m−1 uε,k m ∈ (0, ∞), ε ∈ (0, 1].

And, equivalently, ◦    k¯ m−1 J1 : (a) k¯ m−1 φε,k m (s/ k¯ m−1 uε,k m ) → k¯ m−1 φ0,k m (s) as ε → 0, for s ≥ 0, (b)  k¯ m−1 φ0,k m (·) is the Laplace transform of a non-arithmetic distribution function   (·), (c) k¯ m−1 uε,k ∈ (0, ∞), ε ∈ (0, 1]. without singular component k¯ m−1 F0,k m m  k¯ m−1 K1 :

   (a) k¯ m−1 fε,k < ∞, for ε ∈ (0, 1], (b) k¯ m−1 fε,k / ¯ u → k¯ m−1 f0,k m m k m−1 ε,k m m  < ∞ as ε → 0, (c) k¯ m−1 f0,km is the first moment of the distribution  (·). function k¯ m−1 F0,k m

 The conditions k¯ m−1 L1 and k¯ m−2 LH,1 take the following forms:   k¯ m−1 L1 : k¯ m−1 uε,k m

→

 k¯ m−1 u0,k m

∈ (0, ∞] as ε → 0,

and  k¯ m−2 LH,1 :

Function k¯ m−1 u ·,km belongs to the complete family of asymptotically comparable functions H appearing in condition LH,1 .

Finally, the condition k¯ m−1 R takes the following form: k¯ m−1 R

:

 There exists a function k¯ m−1 q0,k (t, A), t ≥ 0, A ∈ BZ , which belongs to the m  (· k¯ m−1 u ·,km , class P[BZ ], a class of sets Γ ⊆ BZ , and Borel sets U[k¯ m−1 q ·,k m  A)], A ∈ Γ such that: (a) the function k¯ m−1 q0,km (t, A), t ∈ R+, A ∈ BZ is con (t), t ∈ R+ ; (b) the sistent with the tail probability function 1 − k¯ m−1 F0,k m us

   functions k¯ m−1 qε,k (· k¯ m−1 uε,k , A) −→ k¯ m−1 q0,k (·, A) as ε → 0, for points m m m    (· k¯ m−1 u ·,km , A)]) s ∈ U[k¯ m−1 q ·,km (· k¯ m−1 u ·,km , A)], A ∈ Γ; 7 (c) m(U¯ [k¯ m−1 q ·,k m  = 0, for A ∈ Γ; (d) the function k¯ m−1 q˜0,km (·, A) is continuous almost everywhere with respect to the Lebesgue measure m(·) on B+ , for A ∈ Γ.

In this case, the set k¯ m−2 K[k m−1 ] = {k m−1, k m }. Thus, according to Lemma 11.1, the normalisation function,  k¯ m−1 uε,k m

=

k¯ m−1 uε,k m .

(12.188)

The following lemma, which is a direct corollary of Theorem 11.2, takes place. Lemma 12.13 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R, and k¯ m−1 M be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Then: (i) The conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R are satisfied for the shifted regenerative processes (k¯ m−1 ξε(1) (t), k¯ m−1 ηε(1) (t)) in the form of conditions        ◦   k¯ m−1 G1 , k¯ m−1 H1 , k¯ m−1 I1 , k¯ m−1 IH,1 , k¯ m−1 L1 , k¯ m−1 LH,1 , k¯ m−1 J1 (k¯ m−1 J1 ), k¯ m−1 K1 , k¯ m−1 R .

330

12 Ergodic theorems for perturbed MARP

(ii) The quantities and sets appearing in the above conditions can be calculated using the recurrent algorithms described in Lemmas 10.11–10.18, 10.26–10.31, and Theorem 11.2. (iii) The normalisation function uˆ · = k¯ m−1 u ·,km is given by the relation (12.188). 12.2.2.3 Perturbation Conditions for Characteristics Related to Transition Periods of Embedded Regenerative Processes with Partially Removed Virtual Transitions. Let us also analyse conditions connected with the transition period [0, k¯ m−1 τε ). Recall the notation, Dm =

k¯ m−1 X

¯ m = {k1, . . . , k m−1 }. = {k m }, D

(12.189)

The stopping time k¯ m−1 τε is connected with the first hitting time τε,Dm in the domain Dm (for the semi-Markov process ηε (t)) by the following relation: k¯ m−1 τε

= I(ηε (0) ∈

¯ k¯ m−1 X) τε,D m .

(12.190)

Let us introduce the distribution functions, for k n ∈ X,  k¯ m−1 Fε,k n (·)

= Pkn {k¯ m−1 τε ≤ ·}  =

I(· ≥ 0)

for k n ∈ Dm,

¯ m. Pk n {τε,D m ≤ ·} for k n ∈ D

(12.191)

In this case, for k n ∈ X and ε ∈ (0, 1], the hitting probability, k¯ m−1 pε,k n,k m

= Pkn {ηε (k¯ m−1 τε ) = k m } = 1.

(12.192)

It follows from Theorems 10.21 –10.41 and the conditions G1 , H1 , I1 , IH,1 , L1 , ¯ m, LH,1 , J1 , K1 , and k¯ m−1 M that the following relation holds, for k n ∈ D Pk n {τε,D m ≤ · uˇ ε,k n } ⇒ F0,D m,k n k m (·) as ε → 0,

(12.193)

¯ m are given by the relation where: (a) the normalisation functions uˇ ·,kn , k n ∈ D ¯ m are distribution functions concentrated on the (10.115)1 ; (b) F0,Dm,kn km (·), k n ∈ D interval [0, ∞), with Laplace transforms given by relations (10.125)1 and (10.131)1 . ¯ m−1, j ∈ Dm−1 , It is natural to denote, for k n ∈ D  for k n ∈ Dm, j ∈ Dm, 0,kn (·) = I(· ≥ 0) (12.194) ¯k m−1 F ¯ m, j ∈ Dm . F0,Dm,kn km (·) for k n ∈ D The relation (12.20) implies that  k¯ m−1 Fε,k n (· uˇ ε,k n )

= Pkn {k¯ m−1 τε ≤ · uˇε,kn } ⇒ k¯ F0,kn (·) as ε → 0. m−1

(12.195)

12.2 Embedded regenerative processes

331

The relation (12.24) implies that the following condition related to the transition period [0, k¯ m−2 τε ) is satisfied:  k¯ m−1 P2 : k¯ m−1 Fε,k n (· uˇ ε,k n )

⇒

 k¯ m−1 F0,k n (·)

as ε → 0, for n = 1, . . . , m.

It is important that, according to the relation (9.88)1 , m−2 

uˇε,km−1 =

−1 k¯l p¯ε,k m−1 k m−1 uε,k m−1

(12.196)

l=0

and, according to the relation (10.118)1 given in Lemma 10.151 , for any 1 ≤ n ≤ m − 1, uˇε,kn → k¯ m−1 wˇ 0,kn,km−1 ∈ [0, ∞) as ε → 0. (12.197) uˇε,km−1 Recall the sets k¯ n−1 K[k n ] introduced in the relation (11.19). According to this relation, the set k¯ m−1 K[k m ] = {k m−1, k m }. The relations (11.12), (11.20), relations (11.28), (11.29) given in Lemma 11.1 and relation (12.196) imply that uˇε,km−1 = and

 k¯ m−1 uε,k m

Thus, the condition that uˇε,km−1 =  k¯ m−1 uε,k m

k¯ m−2 M

 k¯ m−1 u˜ ε,k m−1  k¯ m−1 u˜ ε,k m

k¯ m−2 u˜ ε,k m−1

=

=

 k¯ m−1 u˜ ε,k m−1

 k¯ m−1 u˜ ε,k m−1 ,

=

k¯ m−2 u˜ ε,k m .

(12.198) (12.199)

and the relation (11.31) given in Lemma 11.1 imply

→

k¯ m−2 w0,k m−1,k m

∈ [0, ∞) as ε → 0.

(12.200)

Lemma 12.14 Let the conditions G1 , H1 , I1 , IH,1 , J1 (J◦1 ), K1 , and k¯ m−1 M be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)) with the transition periods [0, k¯ m−1 τε ). Then: (i) The condition k¯ m−1 P2 is satisfied. (ii) The quantities and sets appearing in the condition k¯ m−1 P2 can be calculated using the recurrent algorithms described in Lemmas 10.131 –10.141 and Theorems 10.21 –10.41 . (iii) The normalisation functions uˇ ·,kn , n = 1, . . . , m − 1, are given by the relation (9.115)1 . (iv Asymptotic relations (12.197) and (12.200) hold. 12.2.2.4 Compressed in Time Embedded Regenerative Processes with Partially Removed Virtual Transitions. Due to the conditions IH,1 and LH,1 , the normalisation functions uˇ ·,kn , 1 ≤ n ≤ m − 1, and k¯ m−1 u ·,km belong to the class of asymptotically comparable functions H. This allows the use, in this case, of the universal normalisation functions,

332

12 Ergodic theorems for perturbed MARP

uˆε =

 k¯ m−1 uε,k m .

(12.201)

Let us consider the multi-alternating regenerative process compressed in time, (ξε, uˆ ε (t), ηε, uˆ ε (t)) = (ξε (t uˆε ), ηε (t uˆε )), t ≥ 0.

(12.202)

The process (ξε, uˆ ε (t), ηε, uˆ ε (t)) can be considered as a multi-alternating regenerative process with a transition period [0, k¯ m−1 τε, uˆ ε ), where  k¯ m−1 τε, uˆ ε

=

k¯ m−1 τε uˆε

.

(12.203)

Let us also consider the corresponding shifted multi-alternating regenerative process compressed in time, (1) (1) (ξε, (t), ηε, (t)) = (ξε, uˆ ε (k¯ m−1 τε, uˆ ε + t), ηε, uˆ ε (k¯ m−1 τε, uˆ ε + t)), t ≥ 0. uˆ  uˆ  ε

ε

(12.204)

The condition

k¯ m−1 W holds for this multi-alternating regenerative process. (1) (t), k¯ m−1 ηε, (t)) be the regenerative process with removed Let, also, (k¯ m−1 ξε,(1) uˆ ε uˆ ε virtual transitions (with the reduced phase space k¯ m−1 X = {k m } of the modulating (1) semi-Markov process k¯ m−1 ηε, (t)) built using the recurrent algorithm for time– uˆ ε

space aggregation of regeneration times (described in Sect. 11.3) applied to the (1) (1) multi-alternating regenerative process (ξε, (t), ηε, (t)). uˆ ε uˆ ε The following relation, which is equivalent to the relation (12.179), takes place, for t ≥ 0, A ∈ BZ and ε ∈ (0, 1], (1) (1) (t, A) = Pi {ξε, (t) ∈ A} = P{k¯ m−1 ξε,(1) (t) ∈ A} Pε, uˆ  uˆ  uˆ  ε

ε

=

ε

Pε(1) (t uˆε ,

A) =

(1) P{ξε (t uˆ ε )

∈ A} = P{k¯ m−1 ξε(1) (t uˆε ) ∈ A}. (12.205)

Relations (12.180)–(12.185) for the multi-alternating regenerative process (ξε (t), ηε (t)) and regenerative process (k¯ m−1 ξε(1) (t), k¯ m−1 ηε(1) (t)) can be re-written for this multi-alternating regenerative process compressed in time with factor uˆε . (1) The probability Pε, (t, A) is, for every A ∈ BZ , a measurable function of t ≥ 0 uˆ ε and the only bounded solution for the following renewal equation, which is an equivalent variant of the renewal equation (12.180), (1)  Pε, (t, A) = k¯ m−1 qε, uˆ ε (t, A) uˆ ε ∫ t (1) + Pε, ((t − s), A) k¯ m−1 Fε, uˆ ε ,km (ds), t ≥ 0, uˆ  ε

0

(12.206)

where, for A ∈ BZ, t ≥ 0,  k¯ m−1 qε, uˆ ε ,i (t,

A) = Pkm {ξε (t uˆε ) ∈ A,

 k¯ m−1 ζε,1

> t uˆε },

(12.207)

12.2 Embedded regenerative processes

and

 k¯ m−1 Fε, uˆ ε ,k m (t)

333  = Pkm {k¯ m−1 ζε,1 ≤ t uˆε }.

(12.208)

Also, the probabilities Pε, uˆ ε ,i (t, A) = Pi {ξε, uˆ ε (t) ∈ A}, i ∈ X are, for every A ∈ BZ , measurable functions of t ≥ 0, which satisfy the following renewal type relations, which is an equivalent variant of the renewal type relations (12.183), Pε, uˆ ε ,i (t, A) = k¯ m−1 qε, uˆ ε ,i (t, A) ∫ t (1) Pε, (t − s, A) k¯ m−1 Fε, uˆ ε ,i (ds), t ≥ 0, i ∈ X, + uˆ  ε

0

(12.209)

where, for A ∈ BZ, t ≥ 0, i ∈ X,  k¯ m−2 qε, uˆ ε ,i (t,

and

A) = Pi {ξε (t uˆε ) ∈ A,

  k¯ m−1 Fε, uˆ ε ,i (t)

k¯ m−2 τε

> t uˆε },

(12.210)

= Pi {k¯ m−1 τε ≤ t uˆε }.

(12.211)

The transition characteristics for the compressed in time regenerative process (1) (k¯ m−1 ξε,(1) (t), k¯ m−1 ηε, (t)) are connected with the transition characteristics that we uˆ  uˆ  ε

ε

have for the regenerative process (k¯ m−1 ξε(1) (t), for ε ∈ (0, 1],

(1) k¯ m−1 ηε (t)) by the following relations,

    k¯ m−1 Q ε, uˆ ε ,k m k m (t) = k¯ m−1 Fε, uˆ ε ,k m (t) = k¯ m−1 Fε,k m (t uˆ ε ), t ≥   k¯ m−1 pε, uˆ ε ,k m k m = k¯ m−1 pε,k m k m = 1,   k¯ m−1 ψε, uˆ ε ,k m k m (s) = k¯ m−1 φε, uˆ ε ,k m (s) ∫ ∞    = k¯ m−1 φε,k (s/uˆε ) = e−su/uˆ ε k¯ m−1 Fε,k (du), s ≥ 0, m m 0   k¯ m−1 eε, uˆ ε ,k m k m = k¯ m−1 fε, uˆ ε ,k m ∫ ∞ −1  −1  = uˆε k¯ m−1 fε,km = uˆε u k¯ m−1 Fε,k (du), m 0    k¯ m−1 qε, uˆ ε ,k m (t, A) = k¯ m−1 qε,k m (t uˆ ε , A), t ≥ 0, A ∈ BZ .

0,

(12.212)

Let us prove that the perturbation conditions formulated in Sects. 1.1.1.1 and (1) (t), k¯ m−1 ηε, (t)). 2.1.2.1 are satisfied for the shifted regenerative processes (k¯ m−1 ξε,(1) uˆ ε uˆ ε Note that the conditions IH,1 and LH,1 imply that uˆε → uˆ0 ∈ (0, ∞] as ε → 0.

(12.213)

The condition k¯ m−1 G1 implies that the condition O1 is satisfied.  Since k¯ m−1 pε, uˆ ε ,k m k m ≡ 1, the condition P1 (a) is automatically satisfied, with the corresponding limiting probability k¯ m−1 p0, uˆ 0,km km = 1.

334

12 Ergodic theorems for perturbed MARP

The conditions k¯ m−1 I1 and k¯ m−1 J1 (k¯ m−1 J1◦ ) imply that the following relation takes place:  k¯ m−1 Fε, uˆ ε ,k m (·)

 k¯ m−1 Q ε,k m k m (· uˆ ε )   ⇒ k¯ m−1 F0,k (·) = k¯ m−1 Q0,k (·) m m km

=

as ε → 0.

(12.214)

The relation (12.170) is equivalent to the following convergence relation for the  (·/uˆε ) of the distribution functions Q ε,km km (· uˆε ), Laplace transforms k¯ m−1 ψε,k m,k m    k¯ m−1 φε, uˆ ε ,k m (s) = k¯ m−1 ψε,k m k m (s/uˆ ε )   → k¯ m−1 φ0,k (s) = k¯ m−1 ψ0,k (s) m m km

as ε → 0, for s ≥ 0.

(12.215)

Note that the condition k¯ m−1 J1 (b) implies that the distribution function k¯ m−1 is weakly non-arithmetic. The above remarks imply that the condition P1 is satisfied for the regenerative (1) processes (k¯ m−1 ξε,(1) (t), k¯ m−1 ηε, (t)). uˆ ε uˆ ε   The conditions k¯ m−1 I1 , k¯ m−1 J1 (k¯ m−1 J1◦ ), and k¯ m−1 K1 imply that the following relation takes place:  F0,u (·) 0,k m

 k¯ m−1 fε, uˆ ε ,k m

  k¯ m−1 eε,k m k m /uˆ ε .   → k¯ m−1 f0,k = k¯ m−1 e0,k m m km

=

as ε → 0.

(12.216)

Taking into account the above remarks, Lemma 10.2 and the relation (12.216), we can conclude that the conditions k¯ m−1 I1 , k¯ m−1 J1 (k¯ m−1 J1◦ ), and k¯ m−1 K1 imply that the

(1) (t), k¯ m−1 ηε, (t)). condition Q1 is satisfied for the regenerative processes (k¯ m−1 ξε,(1) uˆ ε uˆ ε We denote  U[k¯ m−1 q ·, uˆ ,k m (· , A)] = U[k¯ m−1 q ·,k (· k¯ m−1 uˆ ·,km , A)]. m ·

(12.217)

The condition k¯ m−1 R implies that the following relation takes place, for s ∈ U[k¯ m−1 q ·, uˆ ,k (· , A)], m ·

 k¯ m−1 qε, uˆ ε ,k m (·,

A) = us

  k¯ m−1 qε,k m (· k¯ m−1 uˆ ε,k m ,

−→

 k¯ m−1 q0,k m (·,

A)

A) as ε → 0.

(12.218)

The relation (12.218) and the condition k¯ m−1 R (a) imply that the function   k¯ m−1 q0,k m (s, Z) is consistent with the tail probability function 1 − k¯ m−1 F0,k m (s). Also, the relation (12.218) and the condition k¯ m−1 R (d) imply that the func (·, A) is continuous almost everywhere with respect to the Lebesgue tion k¯ m−1 q0,k m measure m(·), for B+ , for A ∈ Γ.

12.2 Embedded regenerative processes

335

Therefore, the conditions k¯ m−1 I1 , k¯ m−1 J1 (k¯ m−1 J1◦ ), k¯ m−1 K1 , and k¯ m−1 R imply that

(1) (t), k¯ m−1 ηε, (t)). condition R1 is satisfied for the regenerative processes (k¯ m−1 ξε,(1) uˆ ε uˆ ε The following lemma summarises the above remarks.

Lemma 12.15 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , R, and k¯ m−1 M be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t))   , k¯ m−1 L1 , k¯ m−1 LH,1 , and, thus, the conditions k¯ m−1 G1 , k¯ m−1 H1 , k¯ m−1 I1 , k¯ m−1 IH,1  ◦   ¯km−1 J1 (k¯ m−1 J1 ), k¯ m−1 K1 , and k¯ m−1 R are satisfied for the regenerative processes (k¯ m−1 ξε(1) (t), k¯ m−1 ηε(1) (t)). Then: (i) The conditions O1 , P1 , Q1 , and R1 are satisfied for the shifted regenerative (1) (t), k¯ m−1 ηε, (t)) with the time compression factor uˆ · = k¯ m−1 u ·,km processes (k¯ m−1 ξε,(1) uˆ ε uˆ ε given by the relation (12.201). (ii) The asymptotic relations (12.214), (12.215), (12.216), and (12.218) play the roles of the asymptotic relations appearing in the above conditions. The corresponding quantities and sets appearing in these conditions are given by the above relations and the relations (12.212) and (12.217).

12.2.2.5 Perturbation Conditions for Characteristics Related to Transition Periods of Compressed in Time Embedded Regenerative Processes with Partially Removed Virtual Transitions. Let us also consider conditions associated with the compressed transition period [0, k¯ m−1 τε, uˆ ε ). The condition k¯ m−1 P2 and the relation (12.200) imply that the following relation holds, for n = 1, . . . , m,   k¯ m−1 Fε,k n (· uˆ ε )

= ⇒

 k¯ m−1 Fε,k n (· uˇ ε,k n

uˆε uˇε,kn

)

−1  k¯ m−1 F0,k n (· k¯ m−2 w0,k m−1,k m )

as ε → 0.

(12.219)

This relation implies that the condition P 1 holds for the multi-alternating regenerative processes (ξε, uˆ ε (t), ηε, uˆ ε (t)) with the transition periods [0, k¯ m−1 τε, uˆ ε ). The above remarks can be summarised in the following lemma. Lemma 12.16 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 (J◦1 ), K1 , and k¯ m−1 M be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Then: (i) The condition P 1 is satisfied for the compressed in time regenerative processes (ξε, uˆ ε (t), ηε, uˆ ε (t)), with the transition periods [0, k¯ m−1 τε, uˆ ε ) and the time compression factor uˆ · = k¯ m−1 u ·,km given by the relation (12.188). (ii) The asymptotic relation (12.219) plays the role of asymptotic relation appearing in the condition P 1 . The corresponding quantities and sets appearing in this condition are given by the relations (12.194), (12.200), and (12.219). Lemmas 12.15 and 12.16 make it possible to apply Theorems 2.1–2.3 to the  regenerative processes (k¯ m−1 ξε, uˆ  (t), k¯ m−1 ηε, (t)), if conditions G1 , H1 , I1 , IH,1 , L1 , uˆ ε ε LH,1 , J1 (J◦1 ), K1 , R, k¯ m−1 M are satisfied.

Chapter 13

Ergodic Theorems for Perturbed Multi-Alternating Regenerative Processes

In this chapter, we present ergodic theorems for perturbed multi-alternating regenerative processes. This chapter includes three sections. In Sect. 13.1, we present ergodic theorems for perturbed multi-alternating regenerative processes based on regularly perturbed embedded alternating regenerative processes with totally and partially removed virtual transitions. The corresponding results are formulated in Theorems 13.1 and 13.2. In Sect. 13.2, we present super-long, long, and short time ergodic theorems for perturbed multi-alternating regenerative processes based on singularly perturbed embedded alternating regenerative processes. The corresponding results are formulated in Theorems 13.3–13.10. In Sect. 13.3, we present ergodic theorems for perturbed multi-alternating regenerative processes based on regularly perturbed embedded regenerative processes. The corresponding results are formulated in Theorems 13.11–13.12. We also clarify the relationship between ergodic theorems for perturbed multi-alternating regenerative processes based on embedded alternating regenerative processes and embedded regenerative processes.

13.1 Ergodic Theorems for Regularly Perturbed Multi-Alternating Regenerative Processes In this section, we present ergodic theorems for perturbed multi-alternating regenerative processes based on regularly perturbed embedded alternating regenerative processes with totally and partially removed virtual transitions.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0_13

337

338

13 Ergodic theorems for perturbed MARP

13.1.1 Ergodic Theorems Based on Regularly Perturbed Embedded Alternating Regenerative Processes with Totally Removed Virtual Transitions 13.1.1.1 Regularly Perturbed Embedded Alternating Regenerative Processes with Totally Removed Virtual Transitions. We shall refer to multi-alternating regenerative processes (ξε (t), ηε (t)) as to regularly perturbed processes in the case, where the corresponding embedded alternating regenerative processes are regularly perturbed. In this subsection, we use the embedded alternating regenerative processes (1) (1) (t), k¯ m−2 ηε,u (t)) based on total removal of virtual transitions for reduced (k¯ m−2 ξε,u ε ε modulating semi-Markov processes. These processes have the phase space Z× k¯ m−2 X, where k¯ m−2 X = {k m−1, k m }. We assume that the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R, and k¯ m−2 M are satisfied. The following two conditions also play an important role: k¯ m−2 Xγ : k¯ m−2 Sβ :

k¯ m−2 u ε, k m ¯ k m−2 u ε, k m−1

→γ=

k¯ m−2 p ε, k m−1 , k m

k¯ m−2 p ε, k m , k m−1

k¯ m−2 γ

→β=

∈ [0, ∞] as ε → 0.

k¯ m−2 β

∈ [0, ∞] as ε → 0.

As mentioned in Sects. 12.1.1.2 and 12.1.14, the conditions G1 , H1 , I1 , IH,1 , L1 , and LH,1 imply that the limits in the asymptotic relations (given in the conditions k¯ m−2 Xγ and k¯ m−2 Sβ ) exist. Moreover, Lemmas 8.21 –8.91 allow calculating these limits. The conditions k¯ m−2 Xγ and k¯ m−2 Sβ are simply separating the cases, where these limits take different values in the interval [0, ∞]. As mentioned in Sect. 12.1.1.2, the conditions G1 , H1 , I1 , IH,1 , L1 , and LH,1 imply that the following condition, which corresponds to regularly perturbed em(1) bedded alternating regenerative processes, is satisfied for the processes (k¯ m−2 ξε,u (t), ε (1) k¯ m−2 ηε,uε (t)):

k¯ m−2 T4 : (a) k¯ m−2 p0,k m k m−1

> 0, or (b) k¯ m−2 p0,km−1 km > 0.

Lemmas 8.21 –8.91 allow calculating the probabilities p0,km−1 km . In the case, where the condition k¯ m−2 T4 is satisfied, β=

k¯ m−2 β

=

k¯ m−2 p0,k m−1 k m k¯ m−2 p0,k m k m−1

.

k¯ m−2 p0,k m k m−1 , k¯ m−2

(13.1)

The condition k¯ m−2 T4 (a) is satisfied if and only if the condition k¯ m−2 T4 is satisfied and the parameter k¯ m−2 β ∈ [0, ∞). Moreover, k¯ m−2 β = 0, if k¯ m−2 p0,km−1 km = 0, while k¯ m−2 β ∈ (0, ∞), if k¯ m−2 p0,k m−1 k m > 0. Similarly, the condition k¯ m−2 T4 (b) is satisfied if and only if the condition k¯ m−2 T4 is satisfied and the parameter k¯ m−2 β ∈ (0, ∞]. Moreover, k¯ m−2 β = ∞, if k¯ m−2 p0,km km−1 = 0, while k¯ m−2 β ∈ (0, ∞), if k¯ m−2 p0,km km−1 > 0.

13.1 Ergodic theorems for regularly perturbed MARP

339

¯ (k¯ p0,km km−1 = k¯ p0,km−1 km = According to Lemma 11.2, the condition k¯ m−2 T m−2 4 m−2 0), which would correspond to singularly perturbed embedded alternating regenerative processes, cannot be satisfied for the alternating regenerative processes (1) (1) (t), k¯ m−2 ηε,u (t)) based on total removal of virtual transitions. (k¯ m−2 ξε,u ε ε 13.1.1.2 Ergodic Theorems Based on Regularly Perturbed Embedded Alternating Regenerative Processes with Totally Removed Virtual Transitions. The parameters β = k¯ m−2 β and γ = k¯ m−2 γ play a key regulatory role in shaping of the corresponding stationary distributions. The corresponding limiting stationary probabilities for perturbed modulating semi-Markov processes are given, for k¯ m−2 β, k¯ m−2 γ ∈ [0, ∞], (k¯ m−2 γ, k¯ m−2 β)  (0, ∞), (∞, 0), by the following relation: ρkm−1 (k¯ m−2 γ, =

(γ) k¯ m−2 e0,k m−1 αk m−1 (k¯ m−2 β) (γ) k¯ m−2 e0,k m−1 αk m−1 (k¯ m−2 β)

ρkm (k¯ m−2 γ, =

k¯ m−2 β)

+

(γ) k¯ m−2 e0,k m αk m (k¯ m−2 β)

,

k¯ m−2 β) (γ) k¯ m−2 e0,k m αk m (k¯ m−2 β)

(γ) k¯ m−2 e0,k m−1 αk m−1 (k¯ m−2 β)

where αkm−1 (k¯ m−2 β) =

1 1+

k¯ m−2 β

+

,

(13.2)

,

(13.3)

(γ) k¯ m−2 e0,k m αk m (k¯ m−2 β)

, αkm (k¯ m−2 β) =

1 1+

k¯ m−2 β

−1

(γ)

and the expectations k¯ m−2 e0,i , i = k m−1, k m are given by the relation (12.55), i.e., (γ) k¯ m−2 e0,k m−1

=

k¯ m−2 e0,k m−1

1+

k¯ m−2 γ

(γ) k¯ m−2 e0,k m

,

=

k¯ m−2 e0,k m . 1 + k¯ m−2 γ −1

(13.4)

(β,γ)

The corresponding limiting stationary probabilities k¯ m−2 π0 (A) for perturbed multi-alternating regenerative processes have the following form, for β = k¯ m−2 β, γ = k¯ m−2 γ ∈ [0, ∞], (k¯ m−2 γ, k¯ m−2 β)  (0, ∞), (∞, 0), and A ∈ BZ ,

(γ,β) (A) = ρ j (k¯ m−2 γ, k¯ m−2 β) k¯ m−2 π0, j (A), (13.5) k¯ m−2 π0 j ∈k¯

m−2

X

where, for A ∈ BZ, j = k m−1, k m , 1 k¯ m−2 π0, j (A) = k¯ m−2 e0, j Note that, for A ∈ BZ ,

∫ 0

∞

k¯ m−2 q0, j (s,

A)m(ds).

(13.6)

340

13 Ergodic theorems for perturbed MARP (0,0) (A) k¯ m−2 π0

and

=

(∞,∞) (A) k¯ m−2 π0

k¯ m−2 π0,k m−1 (A),

=

k¯ m−2 π0,k m (A).

(13.7) (13.8)

Also, recall the corresponding time compression factor uε given by the relations (12.12) and (12.30), that is, uε =

k¯ m−2 uε

=

+

k¯ m−2 uε,k m−1

k¯ m−2 uε,k m ,

(13.9)

where, for i = k m−1, k m , k¯ m−2 uε,i

=

m−3 

−1 k¯r p¯ε,ii uε,i .

(13.10)

r=0

The following theorem takes place. Theorem 13.1 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R, k¯ m−2 M, k¯ m−2 Xγ (for some γ = k¯ m−2 γ ∈ [0, ∞]) and k¯ m−2 Sβ (for some β = k¯ m−2 β ∈ [0, ∞]) be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Let, also, the vector (k¯ m−2 γ, k¯ m−2 β)  (0, ∞), (∞, 0). Then, for any i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0: (i) If γ = k¯ m−2 γ ∈ (0, ∞), β = k¯ m−2 β ∈ [0, ∞], Pε,i (tε k¯ m−2 uε, A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} → (ii) If γ =

k¯ m−2 γ

= 0, β =

(γ,β) (A) k¯ m−2 π0

k¯ m−2 β

as ε → 0.

(13.11)

∈ [0, ∞),

Pε,i (tε k¯ m−2 uε, A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} → (iii) If γ =

k¯ m−2 γ

= ∞, β =

(0,0) (A) k¯ m−2 π0

k¯ m−2 β

as ε → 0.

(13.12)

∈ (0, ∞],

Pε,i (tε k¯ m−2 uε, A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} →

(∞,∞) (A) k¯ m−2 π0

as ε → 0.

(13.13)

Proof Let us prove the statement (i). Lemma 12.3 implies that Theorems 4.2, 4.3, or 4.4 can be applied to obtain the corresponding ergodic relation for the embedded alternating regenerative processes (1) (1) (t), k¯ m−2 ηε,u (t)), if, respectively, k¯ m−2 β ∈ (0, ∞), k¯ m−2 β = 0 or k¯ m−2 β = ∞. (k¯ m−2 ξε,u ε ε By doing this, we obtain that the following ergodic relation, for j ∈ k¯ m−2 X, A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0,

13.1 Ergodic theorems for regularly perturbed MARP

341

(1) (1) Pε,u (tε, A) = P j {k¯ m−2 ξε,u (t ) ∈ A} ε ε ε, j

(1) P j {k¯ m−2 ξε,uε (tε ) ∈ A, = r ∈ k¯

m−2

r ∈ k¯

=

X



→

(1) k¯ m−2 ηε,uε (tε ))

m−2

X

ρr (k¯ m−2 γ,

(γ,β) (A) k¯ m−2 π0

= r}

k¯ m−2 β) k¯ m−2 π0,r (A)

as ε → 0.

(13.14)

The renewal type relation (12.40) can be re-written in the following equivalent form, for i ∈ X, A ∈ Γ and t = tε , Pε,uε ,i (tε, A) = Pi {ξε,uε (tε ) ∈ A} = k¯ m−2 qε,uε ,i (tε, A)

∫ tε (1) Pε,u (tε − s, A) k¯ m−2 Q ε,uε ,i j (ds), + ε, j j ∈ k¯

=

m−2

X

0

k¯ m−2 qε,uε ,i (tε , A)

∫ ∞ (1) + Pε,u (tε ε, j 0 j ∈ k¯ X

− s, A) k¯ m−2 Q ε,uε ,i j (ds)

(13.15)

m−2

(1) where: (a) Pε,u (tε − s, A) = 0, for s > tε and (b) the functions k¯ m−2 qε,uε ,i (t, A) and ε, j the distribution functions Q ε,uε ,i j (·) are given by the relations (12.41) and (12.42). The relation (13.14) implies that the following relation holds, for j ∈ k¯ m−2 X, A ∈ Γ and any 0 ≤ tε → ∞ and 0 ≤ sε → s ∈ [0, ∞) as ε → 0, (1) (tε − sε, A) → Pε,u ε, j

(γ,β) (A) k¯ m−2 π0

as ε → 0.

(13.16)

(1) (tε − s, A), s ≥ 0 converge The above relation means that the functions Pε,u ε, j (γ,β)

locally uniformly to the function P0,(1)j (s, A) = k¯ m−2 π0 (A), s ≥ 0 as ε → 0, in every point s ∈ [0, ∞). Lemma 12.4 implies that, in this case, for i ∈ X, j ∈ k¯ m−2 X, k¯ m−2 pε,i j

= Pi {ηε,uε (k¯ m−2 τε,uε ) = j} →

k¯ m−2 p0,i j

as ε → 0,

(13.17)

and  k¯ m−2 Q ε,uε ,i j (·)

= Pi {k¯ m−2 τε,uε ≤ · , ηε,uε (k¯ m−2 τε,uε ) = j} ⇒ k¯ Q 0,i j (·) as ε → 0, m−2

(13.18)

342

13 Ergodic theorems for perturbed MARP

where the limiting hitting probabilities k¯ m−2 p0,i j = k¯ m−2 Q 0,i j (∞) and the transition probabilities k¯ m−2 Q 0,i j (·) are given, respectively, by the relations (12.62) and (12.64). The relations (13.16), (13.17), and (13.18) imply, by Lemma B.2, that the following relation holds, for i ∈ X, j ∈ k¯ m−2 X, A ∈ Γ and any 0 ≤ tε → ∞, ∫ 0

∞

(1) Pε,u (tε − s, A) k¯ m−2 Q ε,uε ,i j (ds) ε, j ∫ ∞ (β,γ) → (A) k¯ m−2 Q 0,i j (ds) k¯ m−2 π0 0

=

(γ,β) (A) k¯ m−2 p0,i j k¯ m−2 π0

as ε → 0.

(13.19)

Also, the relations (13.17) and (13.18) imply that the following relation holds, for i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0, k¯ m−2 qε,uε ,i (tε ,

A) = Pi {ξε (tε uε ) ∈ A,

k¯ m−2 τε,uε

> tε }

≤ Pi {k¯ m−2 τε,uε > tε } → 0 as ε → 0.

(13.20)

Finally, the relations (13.15), (13.19), and (13.20) imply that the following relation holds, for i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0,

(γ,β) Pε,uε ,i (tε, A) → (A) k¯ m−2 p0,i j k¯ m−2 π0 j ∈ k¯

=

m−2

X

(γ,β) (A) k¯ m−2 π0

as ε → 0.

(13.21)

The proof of the statement (i) is complete. The proofs of the statements (ii) and (iii) are similar. When proving the statement (ii), one should use Theorem 5.1 to obtain the following ergodic relation similar to (13.14): (1) (1) Pε,u (tε, A) = P j {k¯ m−2 ξε,u (t ) ∈ A} ε ε ε, j

→

(0,0) (A) k¯ m−2 π0

as ε → 0.

(13.22)

When proving the statement (iii), one should use Theorem 5.2 to obtain the following ergodic relation similar to (13.14): (1) (1) Pε,u (tε, A) = P j {k¯ m−2 ξε,u (t ) ∈ A} ε ε ε, j

→

(∞,∞) (A) k¯ m−2 π0

as ε → 0.

(13.23)

The rest of the proofs of statements (ii) and (iii) are similar to the corresponding part of the proof of statement (i). 

13.1 Ergodic theorems for regularly perturbed MARP

343

13.1.2 Ergodic Theorems Based on Regularly Perturbed Embedded Alternating Regenerative Processes with Partially Removed Virtual Transitions 13.1.2.1 Regularly Perturbed Embedded Alternating Regenerative Processes with Partially Removed Virtual Transitions. In this subsection, we use the embed(1) (1) ded alternating regenerative processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)) based on partial ε ε removal of virtual transitions for reduced modulating semi-Markov processes. These processes have the phase space Z × k¯ m−2 X, where k¯ m−2 X = {k m−1, k m }. As above, we assume that the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R, and k¯ m−2 M are satisfied. The following two conditions also play an important role:  k¯ m−2 Xγ :

 k¯ m−2 u ε, k m  u ¯ k m−2 ε, k m−1

 k¯ m−2 Sβ :

 k¯ m−2 p ε, k m−1 , k m  k¯ m−2 p ε, k m , k m−1

→ γ =

k¯ m−2 γ

→ β =



∈ [0, ∞] as ε → 0.

k¯ m−2 β



∈ [0, ∞] as ε → 0.

As mentioned in Sects. 12.1.2.2 and 12.1.2.4, the conditions G1 , H1 , I1 , IH,1 , L1 , and LH,1 imply that the limits (in the asymptotic relations given in the conditions   k¯ m−2 Xγ and k¯ m−2 Sβ ) exist. Moreover, Lemmas 8.21 –8.91 allow calculating these limits. The conditions k¯ m−2 Xγ  and k¯ m−2 Sβ  are simply separating the cases, where these limits take different values in the interval [0, ∞]. As mentioned in Sect. 12.1.2.2, the conditions G1 , H1 , I1 , IH,1 , L1 , and LH,1 imply that the following condition, which corresponds to regularly perturbed embedded alternating regenerative processes, can be satisfied for the embedded alternating (1) (1) regenerative processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)): ε

  k¯ m−2 T4 : (a) k¯ m−2 p0,k m k m−1

> 0, or (b)

ε

 k¯ m−2 p0,k m−1 k m

> 0.

Lemmas 8.21 –8.91 allow calculating the probabilities

 p0,k . m−1 k m

 k¯ m−2 p0,k m k m−1 , k¯ m−2

In the case, where the condition k¯ m−2 T4 is satisfied, β =

 k¯ m−2 β =

 k¯ m−2 p0,k m−1 k m  k¯ m−2 p0,k m k m−1

.

(13.24)

The condition k¯ m−2 T4 (a) is satisfied if and only if the condition k¯ m−2 T4 is satisfied  = 0, while and the parameter k¯ m−2 β  ∈ [0, ∞). Moreover, k¯ m−2 β  = 0, if k¯ m−2 p0,k m−1 k m   > 0. ¯k m−2 β ∈ (0, ∞), if k¯ m−2 p0,k m−1 k m Similarly, the condition k¯ m−2 T4 (b) is satisfied if and only if the condition    k¯ m−2 T4 is satisfied and the parameter k¯ m−2 β ∈ (0, ∞]. Moreover, k¯ m−2 β = ∞, if    k¯ m−2 p0,k m k m−1 = 0, while k¯ m−2 β ∈ (0, ∞), if k¯ m−2 p0,k m k m−1 > 0.

344

13 Ergodic theorems for perturbed MARP

  As mentioned in Sect. 16.4.4.3, the condition k¯ m−2 T¯ 4 (k¯ m−2 p0,k = k¯ m−2 m k m−1 = 0), which corresponds to singularly perturbed embedded alternating regenerative processes, can also be satisfied for the embedded alternating regenera(1) (1) tive processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)). ε ε  Conditions separating two cases, when one of the two conditions k¯ m−2 T4 or k¯ m−2 T¯ 4 is satisfied, are given in Remark 11.1. 13.1.2.2 Ergodic Theorems Based on Regularly Perturbed Embedded Alternating Regenerative Processes with Partially Removed Virtual Transitions. In this subsection, we consider the case, where the condition k¯ m−2 T4 is assumed to be satisfied. The parameters β  = k¯ m−2 β  and γ  = k¯ m−2 γ  play a key regulatory role in shaping of the corresponding stationary distributions. The corresponding limiting stationary probabilities for perturbed modulating semi-Markov processes are given, for k¯ m−2 β , k¯ m−2 γ  ∈ [0, ∞], (k¯ m−2 γ , k¯ m−2 β )  (0, ∞), (∞, 0), by the following relation:  p0,k m−1 k m

ρk m−1 (k¯ m−2 γ , =



) (γ )   k¯ m−2 e0,k m−1 αk m−1 (k¯ m−2 β )

(γ)   k¯ m−2 e0,k m−1 αk m−1 (k¯ m−2 β )

ρk m (k¯ m−2 γ , =

k¯ m−2 β

k¯ m−2 β



+

(γ )   k¯ m−2 e0,k m αk m (k¯ m−2 β )

,

) (γ )   k¯ m−2 e0,k m αk m (k¯ m−2 β )

(γ )   k¯ m−2 e0,k m−1 αk m−1 (k¯ m−2 β )

+

(γ )   k¯ m−2 e0,k m αk m (k¯ m−2 β )

,

(13.25)

,

(13.26)

where αk m−1 (k¯ m−2 β ) =

1 1+

k¯ m−2 β



, αk m (k¯ m−2 β ) =

1 1+

k¯ m−2 β

−1

(γ )

and the expectations k¯ m−2 e0,i , i = k m−1, k m are given by the relation (12.117), i.e., (γ ) k¯ m−2 e0,k m−1

=

 k¯ m−2 e0,k m−1 (γ ) , ¯ e 1 + k¯ m−2 γ  km−2 0,km

=

 k¯ m−2 e0,k m . 1 + k¯ m−2 γ −1 (β,γ)

(13.27)

The corresponding limiting stationary probabilities k¯ m−2 π0 (A) for perturbed multi-alternating regenerative processes have the following form, for γ  = k¯ m−2 γ , β  = k¯ m−2 β ∈ [0, ∞], (k¯ m−2 γ , k¯ m−2 β )  (0, ∞), (∞, 0), and A ∈ BZ ,

(γ,β ) (A) = ρj (k¯ m−2 γ , k¯ m−2 β ) k¯ m−2 π0, j (A), (13.28) k¯ m−2 π0 j ∈k¯

m−2

where, for A ∈ BZ, j = k m−1, k m ,

X

13.1 Ergodic theorems for regularly perturbed MARP  k¯ m−2 π0, j (A)

=

∫

1  k¯ m−2 e0, j

∞

0

345

 k¯ m−2 q0, j (s,

A)m(ds).

(13.29)

Note that, for A ∈ BZ,, (0,0) (A) k¯ m−2 π0

and

=

(∞,∞) (A) k¯ m−2 π0

=

 k¯ m−2 π0,k m−1 (A)

(13.30)

 k¯ m−2 π0,k m (A).

(13.31)

Recall also the corresponding time compression factor uε is given by the relation (12.94), that is,   + k¯ m−2 uε,k , (13.32) uε = k¯ m−2 uε = k¯ m−2 uε,k m m−1  where the functions k¯ m−2 uε,k , ¯ u are given by the relations (11.19)–(11.21). m−1 k m−2 ε,k m The following theorem takes place.

Theorem 13.2 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R, and k¯ m−2 M,        k¯ m−2 T4 , k¯ m−2 Xγ (for some γ = k¯ m−2 γ ∈ [0, ∞]), and k¯ m−2 Sβ (for some β = k¯ m−2 β ∈ [0, ∞]) be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Let, also, the vector (k¯ m−2 γ , k¯ m−2 β )  (0, ∞), (∞, 0). Then, for any i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0: (i) If γ  = k¯ m−2 γ  ∈ (0, ∞), β  = k¯ m−2 β  ∈ [0, ∞], Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} → (ii) If γ  =

k¯ m−2 γ



= 0, β  =

(γ,β ) (A) k¯ m−2 π0

k¯ m−2 β



as ε → 0.

(13.33)

∈ [0, ∞),

Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} → (iii) If γ  =

k¯ m−2 γ



= ∞, β  =

(0,0) (A) k¯ m−2 π0

k¯ m−2 β



as ε → 0.

(13.34)

∈ (0, ∞],

Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} →

(∞,∞) (A) k¯ m−2 π0

as ε → 0.

(13.35)

Proof The proof of the statement (i) is similar to the proof of the statement (i) in Theorem 13.1. In this case, the statements (i)–(iii) of Lemma 12.7 imply that Theorems 4.2, 4.3, or 4.4 can be applied to obtain the ergodic relation similar to (13.14) for the embed(1) (1) ded alternating regenerative processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)), if, respectively, ε ε    k¯ m−2 β ∈ (0, ∞), k¯ m−2 β = 0 or k¯ m−2 β = ∞.

346

13 Ergodic theorems for perturbed MARP

Also, Lemma 12.8 is used, instead of Lemma 12.4, to obtain relations similar to (13.19) and (13.20) for the embedded alternating regenerative processes (1) (1) (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)). ε ε Finally, the above-mentioned relations imply that a relation similar to (13.21) (1) (1) holds for the embedded alternating regenerative processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)). ε ε When proving the statement (ii), one should use Theorem 5.1 to obtain an ergodic relation similar to (13.22) for the embedded alternating regenerative processes (1) (1) (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)). ε ε When proving the statement (iii), one should use Theorem 5.2 to obtain an ergodic relation similar to (13.23) for the embedded alternating regenerative processes (1) (1) (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)). ε ε The rest of the proofs of statements (ii) and (iii) are similar to the corresponding part of the proof of statement (i). 

13.2 Ergodic Theorems for Singularly Perturbed Multi-Alternating Regenerative Processes In this section, we present super-long, long, and short time ergodic theorems for perturbed multi-alternating regenerative processes based on singularly perturbed embedded alternating regenerative processes.

13.2.1 Super-Long and Long Time Ergodic Theorems Based on Singularly Perturbed Embedded Alternating Regenerative Processes 13.2.1.1 Singularly Perturbed Embedded Alternating Regenerative Processes with Partially Removed Virtual Transitions. In this subsection, we again use the (1) (1) embedded alternating regenerative processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)). ε ε As in Sect. 13.1.2, we assume that the conditionsG1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R and k¯ m−2 M, k¯ m−2 Xγ  , k¯ m−2 Sβ  are satisfied. However, we now assume that the embedded alternating regenerative processes (1) (1) (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)) are singularly perturbed, i.e., the following condition is ε ε satisfied:   ¯ k¯ m−2 T4 : k¯ m−2 p0,k m k m−1 , k¯ m−2 p0,k m−1 k m

= 0.

Let us introduce the time compression factor vε defined by the following relation: vε =

 k¯ m−2 vε

=

−1 k¯ m−2 p¯ε,k m−1 k m−1

+

−1 k¯ m−2 p¯ε,k m k m ,

ε ∈ (0, 1].

(13.36)

13.2 Ergodic theorems for singularly perturbed MARP

347

Note also that the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , and k¯ m−2 M imply, by The  → k¯ m−2 p¯0,k orem 11.2, that the transition probabilities, 0 < k¯ m−2 p¯ε,k m−1 k m−1 m−1 k m−1   and 0 < k¯ m−2 p¯ε,km km → k¯ m−2 p¯0,km km as ε → 0. ¯ , Thus, by the condition k¯ m−2 T 4  k¯ m−2 vε

→ ∞ as ε → 0.

(13.37)

13.2.1.2 Super-Long Time Ergodic Theorems Based on Singularly Perturbed Embedded Alternating Regenerative Processes. Let us describe the asymptotic behaviour of the probabilities Pε,i (tε uε , A) for the so-called super-long times 0 ≤ tε → ∞ as ε → 0, which satisfy the following asymptotic relation: tε

 k¯ m−2 vε

→ ∞ as ε → 0.

(13.38)

The corresponding limiting stationary probabilities for perturbed multi-alternating regenerative processes are given by the relations (13.25)–(13.31). We also recall the corresponding time compression factor uε = k¯ m−2 uε given by the relation (13.32). The following theorem, which is an analogue of Theorem 13.2 for singularly perturbed multi-alternating regenerative processes, takes place. Theorem 13.3 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R, and k¯ m−2 M,     ¯ k¯ m−2 T4 , k¯ m−2 Xγ (for some γ = k¯ m−2 γ ∈ [0, ∞]), and k¯ m−2 Sβ (for some β = k¯ m−2 β ∈ [0, ∞]) be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Let, also, vector (k¯ m−2 γ , k¯ m−2 β )  (0, ∞), (∞, 0). Then, for any A ∈ Γ, i ∈ X and 0 ≤ tε → ∞ as ε → 0 such that tε / k¯ m−2 vε → ∞ as ε → 0: (i) If γ  = k¯ m−2 γ  ∈ (0, ∞), β  = k¯ m−2 β  ∈ [0, ∞], Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} → (ii) If γ  =

k¯ m−2 γ



= 0, β  =

(γ,β ) (A) k¯ m−2 π0

k¯ m−2 β



as ε → 0.

(13.39)

∈ [0, ∞),

Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} → (iii) If γ  =

k¯ m−2 γ



= ∞, β  =

(0,0) (A) k¯ m−2 π0

k¯ m−2 β



as ε → 0.

(13.40)

∈ (0, ∞],

Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} →

(∞,∞) (A) k¯ m−2 π0

as ε → 0.

(13.41)

Proof The proof is analogous to the proof of Theorem 13.1. In this case, the statements (iv)–(vi) of Lemma 12.7 imply that Theorem 6.1 (for the case, where

348

13 Ergodic theorems for perturbed MARP

γ  ∈ (0, ∞)), 8.1 (γ  = 0), or 8.2 (γ  = ∞) can be used to obtain an ergodic relation similar to, respectively, (13.14), (13.22), or (13.23), for the embedded alternating (1) (1) regenerative processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)). ε ε Also, Lemma 12.8 can be used to obtain relations similar to (13.19) and (13.20), (1) (1) for the embedded alternating regenerative processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)). ε ε Finally, the above-mentioned relations imply that a relation similar to (13.21) (1) holds for the embedded alternating regenerative processes (k¯ m−2 ξε,u  (t), ε

(1) k¯ m−2 ηε,uε (t)).



Remark 13.1 The difference between Theorems 13.2 and 13.3 creates the condition  k¯ m−2 T4 , which corresponds to a simpler model with regular perturbations, and the  condition k¯ m−2 T¯ 4 , which corresponds to the more complex model with singular perturbations. In Theorem 13.2, the corresponding ergodic relation for the probabilities Pε,i (tε k¯ m−2 uε , A) holds for any 0 ≤ tε → ∞ as ε → 0. In Theorem 13.3, the corresponding ergodic relation for the probabilities Pε,i (tε k¯ m−2 uε , A) holds for any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → ∞ as ε → 0. This means that in the latter case, an additional condition is imposed on the rate of convergence to ∞ for time tε k¯ m−2 uε since, vε → ∞ as ε → 0. 13.2.1.3 Long Time Ergodic Theorems Based on Singularly Perturbed Embedded Alternating Regenerative Processes. Let us describe the asymptotic behaviour of the probabilities Pε,i (tε uε , A) for the so-called long times 0 ≤ tε → ∞ as ε → 0, which satisfy the following asymptotic relation: tε

 k¯ m−2 vε

→ t ∈ (0, ∞) as ε → 0.

(13.42)

The corresponding limiting stationary probabilities for perturbed multi-alternating regenerative processes are given by the relations (13.30)–(13.31). As above, the corresponding time compression factor uε = k¯ m−2 uε is given by the relation (13.32). The following theorem takes place. Theorem 13.4 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R, and k¯ m−2 M,       ¯ k¯ m−2 T4 , k¯ m−2 Xγ (for some γ = k¯ m−2 γ ∈ [0, ∞]), and k¯ m−2 Sβ (for some β = k¯ m−2 β ∈ [0, ∞]) be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Let, also, vector (k¯ m−2 γ , k¯ m−2 β )  (0, ∞), (∞, 0). Then, for any i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0 such that tε /vε → t ∈ (0, ∞) as ε → 0: (i) If k¯ m−2 γ  ∈ (0, ∞), k¯ m−2 β  = 0, Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} → (ii) If k¯ m−2 γ  ∈ (0, ∞),

k¯ m−2 β



(0,0) (A) k¯ m−2 π0

= ∞,

as ε → 0.

(13.43)

13.2 Ergodic theorems for singularly perturbed MARP

349

Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} → (iii) If k¯ m−2 γ  = 0,

k¯ m−2 β



(∞,∞) (A) k¯ m−2 π0

as ε → 0.

(13.44)

∈ [0, ∞),

Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} → (iv) If k¯ m−2 γ  = ∞,

k¯ m−2 β



(0,0) (A) k¯ m−2 π0

as ε → 0.

(13.45)

∈ (0, ∞],

Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} →

(∞,∞) (A) k¯ m−2 π0

as ε → 0.

(13.46)

Proof The proof is similar to the proof of Theorem 13.1. In this case, the statements (iv)–(vi) of Lemma 12.7 imply that Theorem 6.2 (for the cases, where γ  ∈ (0, ∞)), 8.1 (γ  = 0), or 8.2 (γ  = ∞) can be applied to obtain an ergodic relation similar to, respectively, (13.14), (13.22), or (13.23), for the embedded alter(1) (1) nating regenerative processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)). ε ε Also, Lemma 12.8 can be used to obtain relations similar to (13.19) and (13.20), (1) (1) for the embedded alternating regenerative processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)). ε ε Finally, the above-mentioned relations imply that a relation similar to (13.21) (1) holds for the embedded alternating regenerative processes (k¯ m−2 ξε,u  (t), ε

(1) k¯ m−2 ηε,u (t)). ε



The most interesting is the case where parameters γ = 

k¯ m−2 γ



, β =

k¯ m−2 β



∈ (0, ∞).

(13.47)



Let η(γ ,β ) (t), t ≥ 0 be a homogeneous continuous time Markov chain with the phase space k¯ m−2 X = {k m−1, k m }, the transition probabilities of embedded Markov chain p jr = I( j  r), j, r ∈ k¯ m−2 X, and the distribution functions of so(γ,β )

journ times in states k m−1 and k m , respectively, k¯ m−2 Fkm−1 (t) = 1 − exp{−t(1 +k¯ m−2 (γ )

(γ,β )

β )/ k¯ m−2 e0,km−1 }, t ≥ 0 and k¯ m−2 Fkm (γ )

(γ )

(t) = 1−exp{−t(1+ k¯ m−2 β −1 )/ k¯ m−2 e0,km }, t ≥

0, where expectations e0,i , i = k m−1, k m , are given by relation (12.117). We also assume that this Markov chain is continuous from the right trajectories.   (γ,β ) Let p jr (t) = P j {η(γ ,β ) (t) = r }, t ≥ 0, j, r ∈ k¯ m−2 X be the transition probabili  ties for the Markov chain η(γ ,β ) (t). (γ,β ) Explicit expressions for the transition probabilities p jr (t) are well known as the solutions of the corresponding forward Kolmogorov system of differential equations (γ,β ) for these probabilities. Namely, the corresponding 2 × 2 matrix p jr (t) has the following form, for t ≥ 0,

350

13 Ergodic theorems for perturbed MARP (γ , β  )

k¯ m−2 p j r

(t) 

     ρ (γ, β ) + ρk m (γ, β )e−λ(γ , β )t ρk m (γ, β ) − ρk m (γ, β )e−λ(γ , β )t =  k m−1   , β  )t , β  )t   −λ(γ     −λ(γ ρk m−1 (γ , β ) − ρk m−1 (γ , β )e ρk m (γ , β ) + ρk m−1 (γ , β )e

  , 

(13.48)

where λkm−1 (γ , β ) =





λkm (γ , β ) =

λ(β , γ ) =



⎧ ⎪ ⎨ ⎪

β m−2 (γ  ) k¯ m−2 e0, k m−1 1+ k¯

for k¯ m−2 β  ∈ [0, ∞),

⎪ ⎪∞ ⎩ ⎧ ⎪ ⎨∞ ⎪ ⎪ ⎪ ⎩

β−1

1+ k¯

m−2 (γ  ) k¯ m−2 e0, k m

for k¯ m−2 β  = ∞, for k¯ m−2 β  = 0,

(13.49)

for k¯ m−2 β  ∈ (0, ∞].

λkm−1 (γ , β ) + λkm (γ , β ) for k¯ m−2 β  ∈ (0, ∞), ∞ for k¯ m−2 β  = 0, ∞,

(13.50)

and

ρkm−1 (γ , β ) =

ρkm (γ , β ) =

⎧ 1 ⎪ ⎪ ⎪ ⎨λ ⎪ ⎪ ⎪ ⎪ ⎪0 ⎩

⎧ 0 ⎪ ⎪ ⎪ ⎨ λk ⎪ ⎪ ⎪ ⎪ ⎪1 ⎩

  k m (γ ,β ) λ(γ,β )

(γ,β )

m−1 λ(γ,β )

=

=

for k¯ m−2 β = 0,

(γ  )  −1 k¯ m−2 e0, k m−1 (1+ k¯ m−2 β ) (γ  ) k¯ m−2 e0

for k¯ m−2 β  ∈ (0, ∞), for k¯ m−2 β  = ∞, for k¯ m−2 β  = 0,

(γ  ) −1 )−1 k¯ m−2 e0, k m (1+ k¯ m−2 β (γ  ) k¯ m−2 e0

for k¯ m−2 β  ∈ (0, ∞), for k¯ m−2 β  = ∞,

(13.51)

where (γ,β ) k¯ m−2 e0

(γ ) k¯ m−2 e0,k m−1 1 + k¯ m−2 β 

=

+

(γ ) k¯ m−2 e0,k m

1+

k¯ m−2 β

−1

,

(13.52)

(γ )

and the expectations e0, j , j = k m−1, k m are given by the relation (13.27), i.e., (γ ) k¯ m−2 e0,k m−1

=

 k¯ m−2 e0,k m−1 (γ ) , k¯ m−2 e0,km  1 + k¯ m−2 γ 



=

 k¯ m−2 e0,k m . 1 + k¯ m−2 γ −1

Note that the Markov chain η(β ,γ ) (t) is ergodic and ρi (β , γ ), i ∈ stationary probabilities.

(13.53) k¯ m−2 X,

are its

13.2 Ergodic theorems for singularly perturbed MARP

351

The corresponding limiting stationary probabilities for the singularly perturbed multi-alternating regenerative processes ξε (t) have the following form, for k¯ m−2 γ ,  k¯ m−2 β ∈ (0, ∞), t ∈ (0, ∞), i ∈ X, and A ∈ BZ ,

(γ,β ) (γ,β ) (t, A) = (t, A), (13.54) k¯ m−2 p0,i j k¯ m−2 π0, jr k¯ m−2 π0,i j,r ∈ k¯

where, for A ∈ BZ, j, r ∈

k¯ m−2 X

m−2

X

and t ∈ (0, ∞),

(γ,β ) (t, k¯ m−2 π0, jr

A) =

(γ,β )  (t) k¯ m−2 π0,r (A), k¯ m−2 p jr

(13.55)

 (A), r = k and the limiting stationary probabilities k¯ m−2 π0,r m−1, k m, A ∈ BZ are given by the relation (13.29), and the limiting hitting probabilities k¯ m−2 p0,i j for the corresponding transition period are given by the relation (12.128). As above, the corresponding time compression factor uε = k¯ m−2 uε is given by the relation (13.32). The following theorem takes place.

Theorem 13.5 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R, and k¯ m−2 M,       ¯ k¯ m−2 T4 , k¯ m−2 Xγ (for some γ = k¯ m−2 γ ∈ (0, ∞)), and k¯ m−2 Sβ (for some β = k¯ m−2 β ∈ (0, ∞)) be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Then, for any i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0 such that tε /vε → t ∈ (0, ∞) as ε → 0, Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} →

(γ,β ) (t, k¯ m−2 π0,i

A) as ε → 0.

(13.56)

Proof The statement (iv) of Lemma 12.7 implies that Theorem 6.2 can be applied to obtain the corresponding ergodic relation for the embedded alternating regenerative (1) (1) processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)). ε ε By doing this, we obtain the following ergodic relation, for j, r ∈ k¯ m−2 X, A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → t ∈ (0, ∞) as ε → 0, (1) (1) Pε,u  (tε , A) = P j {k¯ m−2 ξε,u  (tε ) ∈ A} ε, j ε

(1) = P j {k¯ m−2 ξε,u (tε ) ∈ A, r ∈ k¯

→

m−2

X



r ∈ k¯

m−2

X

ε

(γ,β ) (t, k¯ m−2 π0, jr

(1) k¯ m−2 ηε,uε (tε ))

A) as ε → 0.

= r} (13.57)

The renewal type relation (12.104) can be re-written in the following equivalent form, for i ∈ X, A ∈ Γ and t = tε ,

352

13 Ergodic theorems for perturbed MARP

Pε,uε ,i (tε, A) = Pi {ξε,uε (tε ) ∈ A} = k¯ m−2 qε,uε ,i (tε, A)

∫ tε (1)   ,i j (ds), Pε,u +  , j (tε − s, A) k¯ m−2 Q ε,u ε j ∈ k¯

=

m−2

X

ε

0

 k¯ m−2 qε,uε ,i (tε , A)

∫ ∞ (1) + Pε,u  (tε ε, j 0 j ∈ k¯ X

− s, A) k¯ m−2 Q ε,uε ,i j (ds)

(13.58)

m−2

(1) ε,uε ,i (t, A) and where: (a) Pε,u  (tε − s, A) = 0, for s > tε and (b) the functions k¯ m−2 q ε, j the distribution functions Q ε,uε ,i j (·) are given by the relations (12.105) and (12.106). The relation (13.57) implies that the following relation holds, for j, r ∈ k¯ m−2 X, A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → t ∈ (0, ∞) as ε → 0 and 0 ≤ sε → s ∈ [0, ∞) as ε → 0,

(γ,β ) (1) Pε,u (t, A) as ε → 0. (13.59)  , j (tε − sε , A) → k¯ m−2 π0, jr ε

r ∈ k¯

m−2

X

Indeed, in this case, (tε − sε )/vε → t as ε → 0 and, thus, the relation (13.59) is implied by the relation (13.57). (1) The above relation means that the functions Pε,u  (tε − s, A), s ≥ 0 converge ε, j  (γ,β ) (1) locally uniformly to the function π0, j (s, t, A) = r ∈ k¯ X k¯ m−2 π0, jr (t, A), s ≥ 0 as m−2 ε → 0 in any point s ∈ [0, ∞). Lemma 12.8 implies that, in this case, for i ∈ X, j ∈ k¯ m−2 X, k¯ m−2 pε,i j

= Pi {ηε,uε (k¯ m−2 τε,uε ) = j} →

k¯ m−2 p0,i j

as ε → 0,

(13.60)

and   k¯ m−2 Q ε,uε ,i j (·)

= Pi {k¯ m−2 τε,uε ≤ ·, ηε,uε (k¯ m−2 τε,uε ) = j} ⇒ k¯ Q  (·) as ε → 0, m−2

0,i j

(13.61)

 (∞) and the transition where the limiting hitting probabilities k¯ m−2 p0,i j =k¯ m−2 Q 0,i j  (·) are given by the relations (12.128) and (12.130), respecprobabilities k¯ m−2 Q 0,i j tively. The relations (13.59), (13.60), and (13.61) imply, by Lemma B.2, that the following relation holds, for i ∈ X, j ∈ k¯ m−2 X, A ∈ Γ and any 0 ≤ tε → ∞ such that tε /vε → t ∈ (0, ∞) as ε → 0,

13.2 Ergodic theorems for singularly perturbed MARP

∫ 0

∞

353

(1)   ,i j (ds) Pε,u  (tε − s, A) k¯ m−2 Q ε,u ε ε, j ∫ ∞ → π0,(1)j (s, t, A) k¯ m−2 Q 0,i j (ds) 0

(γ,β ) (t, A) k¯ m−2 p0,i j as ε → 0. = k¯ m−2 π0, jr r ∈ k¯

m−2

(13.62)

X

Also, the relation (13.61) implies that the following relation holds, for i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0, such that tε /vε → t ∈ (0, ∞) as ε → 0,  k¯ m−2 qε,uε ,i (tε ,

A) = Pi {ξε (tε uε ) ∈ A,

 k¯ m−2 τε,uε

> tε }

≤ Pi {k¯ m−2 τε,uε > tε } → 0 as ε → 0.

(13.63)

Finally, the relations (13.58), (13.62), and (13.63) imply that the following relation holds, for i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0, such that tε /vε → t ∈ (0, ∞) as ε → 0,



(γ,β ) Pε,uε ,i (tε, A) → (t, A) p0,i j k¯ m−2 π0, jr j ∈ k¯

=

m−2

X r ∈ k¯

(γ,β )

k¯ m−2 π0,i

m−2

X

(t, A) as ε → 0.

(13.64)

The proof is complete.



13.2.2 Short Time Ergodic Theorems Based on Singularly Perturbed Embedded Alternating Regenerative Processes 13.2.2.1 Short Time Ergodic Theorems for Singularly Perturbed MultiAlternating Regenerative Processes Based on the First Time Compression Factor. Let us describe the asymptotic behaviour of the probabilities Pε,i (tε uε , A) for the so-called short times 0 ≤ tε → ∞ as ε → 0, which satisfy the following asymptotic relation: tε (13.65)  → 0 as ε → 0. k¯ m−2 vε  We also assume that the conditions k¯ m−2 T¯ 4 , k¯ m−2 Xγ  (for some γ  = k¯ m−2 γ  ∈ (0, ∞)), and k¯ m−2 Sβ (for some β  = k¯ m−2 β  ∈ (0, ∞)) are satisfied. The corresponding limiting stationary probabilities for perturbed multi-alternating regenerative processes have the following form, for γ  = k¯ m−2 γ , β  = k¯ m−2 β  ∈ (0, ∞), i ∈ X, and A ∈ BZ ,

354

13 Ergodic theorems for perturbed MARP  k¯ m−2 π¯ 0,i (A)



=

j,r ∈ k¯

m−2

X

 k¯ m−2 p0,i j k¯ m−2 π¯ 0, jr (A),

(13.66)

where  k¯ m−2 π¯ 0, jr (A)

 = I(r = j) k¯ m−2 π0,r (A)

 =

 k¯ m−2 π0,r (A)

0

for A ∈ BZ, r = j, j, r ∈ for A ∈ BZ, r  j, j, r ∈

k¯ m−2 X,

(13.67)

k¯ m−2 X

 (A), are given by the relation (13.29), and the limiting stationary probabilities k¯ m−2 π0,r and the limiting hitting probabilities k¯ m−2 p0,i j are given by the relation (12.128). As above, the corresponding time compression factor uε = k¯ m−2 uε is given by the relation (13.32). The following theorem takes place.

Theorem 13.6 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R, and k¯ m−2 M,       ¯ k¯ m−2 T4 , k¯ m−2 Xγ (for some γ = k¯ m−2 γ ∈ (0, ∞)), and k¯ m−2 Sβ (for some β = k¯ m−2 β ∈ (0, ∞)) be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Then, for any i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0 such that tε /vε → 0 as ε → 0, Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} →

 k¯ m−2 π¯ 0,i (A)

as ε → 0.

(13.68)

Proof It is similar to the proof of Theorem 13.5. The statement (iv) of Lemma 12.7 implies that Theorem 7.1 can be applied to obtain the corresponding ergodic relation (1) (1) for the embedded alternating regenerative processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)). ε ε By doing this, we obtain the following ergodic relation, for j, r ∈ k¯ m−2 X, A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → 0 as ε → 0, (1) (1) Pε,u  (tε , A) = P j {k¯ m−2 ξε,u  (tε ) ∈ A} ε, j ε

(1) = P j {k¯ m−2 ξε,u (tε ) ∈ A, r ∈ k¯

→

m−2

X



r ∈ k¯

m−2

X

ε

 k¯ m−2 π¯ 0, jr (A)

(1) k¯ m−2 ηε,uε (tε ))

as ε → 0.

= r} (13.69)

The renewal type relation (12.104) can be re-written in the following equivalent form, for i ∈ X, A ∈ Γ and t = tε ,

13.2 Ergodic theorems for singularly perturbed MARP

355

Pε,uε ,i (tε, A) = Pi {ξε,uε (tε ) ∈ A} = k¯ m−2 qε,uε ,i (tε, A)

∫ tε (1)   ,i j (ds), Pε,u +  , j (tε − s, A) k¯ m−2 Q ε,u ε j ∈ k¯

=

m−2

X

ε

0

 k¯ m−2 qε,uε ,i (tε , A)

∫ ∞ (1) + Pε,u  (tε ε, j 0 j ∈ k¯ X

− s, A) k¯ m−2 Q ε,uε ,i j (ds)

(13.70)

m−2

(1) ε,uε ,i (t, A) and where: (a) Pε,u  (tε − s, A) = 0, for s > tε and (b) the functions k¯ m−2 q ε, j the distribution functions Q ε,uε ,i j (·) are given by the relations (12.105) and (12.106). The relation (13.69) implies that the following relation holds, for j, r ∈ k¯ m−2 X, A ∈ Γ and any 0 ≤ tε → ∞ as ε → 0 such that tε /vε → 0 as ε → 0 and 0 ≤ sε → s ∈ [0, ∞) as ε → 0,

(1)  (13.71) Pε,u  , j (tε − sε , A) → k¯ m−2 π¯ 0, jr (A) as ε → 0. ε

r ∈ k¯

m−2

X

Indeed, in this case, (tε − sε )/vε → 0 as ε → 0 and, thus, the relation (13.71) is implied by the relation (13.69). (1) The above relation means that the functions Pε,u  (tε − s, A), s ≥ 0 converge ε, j  (1) locally uniformly to the function π¯0, j (s, A) = r ∈ k¯ X k¯ m−2 π¯0, jr (A), s ≥ 0 as ε → 0 m−2 in any point s ∈ [0, ∞). Lemma 12.8 implies that, in this case, for i ∈ X, j ∈ k¯ m−2 X, k¯ m−2 pε,i j

= Pi {ηε,uε (k¯ m−2 τε,uε ) = j} →

k¯ m−2 p0,i j

as ε → 0,

(13.72)

and   k¯ m−2 Q ε,uε ,i j (·)

= Pi {k¯ m−2 τε,uε ≤ ·, ηε,uε (k¯ m−2 τε,uε ) = j} ⇒ k¯ Q  (·) as ε → 0, m−2

0,i j

(13.73)

 (∞) and the transition where the limiting hitting probabilities k¯ m−2 p0,i j =k¯ m−2 Q 0,i j  probabilities k¯ m−2 Q 0,i j (·) are given by the relations (12.128) and (12.130), respectively. Relations (13.71), (13.72), and (13.73) imply, by Lemma B.2, that the following relation holds, for i ∈ X, j ∈ k¯ m−2 X, A ∈ Γ and any 0 ≤ tε → ∞ such that tε /vε → 0 as ε → 0,

356

13 Ergodic theorems for perturbed MARP

∫

∞

0

(1)   ,i j (ds) Pε,u  (tε − s, A) k¯ m−2 Q ε,u ε ε, j ∫ ∞ → π¯0,(1)j (s, A) k¯ m−2 Q 0,i j (ds) 0

 = k¯ m−2 π¯ 0, jr (A) k¯ m−2 p0,i j as ε → 0. r ∈ k¯

m−2

(13.74)

X

Also, the relation (13.73) obviously implies that the following relation holds, for i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0, such that tε /vε → 0 as ε → 0,  k¯ m−2 qε,uε ,i (tε ,

A) = Pi {ξε (tε uε ) ∈ A,

 k¯ m−2 τε,uε

> tε }

≤ Pi {k¯ m−2 τε,uε > tε } → 0 as ε → 0.

(13.75)

Finally, the relations (13.70), (13.74), and (13.75) imply that the following relation holds, for i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0, such that tε /vε → 0 as ε → 0,



 Pε,uε ,i (tε, A) → k¯ m−2 π¯ 0, jr (A) p0,i j j ∈ k¯

=

m−2

X r ∈ k¯

 k¯ m−2 π¯ 0,i (A)

m−2

X

as ε → 0.

(13.76) 

The proof is complete.

13.2.2.2 First Type Short Time Ergodic Theorems for Singularly Perturbed Multi-Alternating Regenerative Processes Based on the Second Time Compression Factor. Let us introduce the second time compression factor wε defined by the following relation: wε =

 k¯ m−2 wε

 = (k¯ m−2 p¯ε,k + m−1 k m−1

 −1 k¯ m−2 p¯ε,k m k m ) , ε

∈ (0, 1].

(13.77)



The condition k¯ m−2 T¯ 4 implies that  k¯ m−2 wε

→ ∞ as ε → 0. 

According to Lemma 7.1, if the condition k¯ m−2 T¯ 4 and the condition  k¯ m−2 S∞ hold, then   k¯ m−2 wε ≺ k¯ m−2 vε as ε → 0.

(13.78)  k¯ m−2 S0

or

(13.79)

In this subsection, we describe the asymptotic behaviour of the probabilities   Pε,i (tε , A) for the case, where the conditions k¯ m−2 T¯ 4 and k¯ m−2 S0 or k¯ m−2 S∞ are satisfied, and 0 ≤ tε → ∞ as ε → 0 such that  k¯ m−2 wε

≺ tε ≺

 k¯ m−2 vε

as ε → 0.

(13.80)

The corresponding stationary probabilities for perturbed multi-alternating regenerative processes take the forms given in the relations (13.30)–(13.31).

13.2 Ergodic theorems for singularly perturbed MARP

357

As above, the corresponding time compression factor uε = the relation (13.32). The following theorem takes place.

 k¯ m−2 uε

is given by

Theorem 13.7 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R, and k¯ m−2 M,       ¯ k¯ m−2 T4 , k¯ m−2 Xγ (for some γ = k¯ m−2 γ ∈ [0, ∞]), and k¯ m−2 Sβ (for β = 0 or β = ∞) be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Let, also, vector (k¯ m−2 γ , k¯ m−2 β )  (0, ∞), (∞, 0). Then, for any i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0 such that tε /vε → 0 as ε → 0 and tε /wε → ∞ as ε → 0: (i) If k¯ m−2 γ  ∈ [0, ∞) k¯ m−2 β  = 0, Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} →

(0,0) (A) k¯ m−2 π0

as ε → 0.

(13.81)

(ii) If k¯ m−2 γ  ∈ (0, ∞] k¯ m−2 β  = ∞, Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} →

(∞,∞) (A) k¯ m−2 π0

as ε → 0.

(13.82)

Proof The proof is similar to the proof of Theorem 13.1. In this case, the statements (iv)–(vi) of Lemma 12.7 imply that Theorems 7.2, 7.3 (for the cases, where γ  ∈ (0, ∞)), 8.3 (γ  = 0) or 8.4 γ  = ∞) can be applied to obtain an ergodic relation similar to, respectively, (13.14), (13.22), or (13.23), for the embedded alternating (1) (1) regenerative processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)). ε ε Also, Lemma 12.8 can be used to obtain relations similar to (13.19) and (13.20), (1) (1) for the embedded alternating regenerative processes (k¯ m−2 ξε,u  (t), k¯ m−2 ηε,u  (t)). ε ε Finally, the above-mentioned relations imply that a relation similar to the relation (1) (13.21) holds for the embedded alternating regenerative processes (k¯ m−2 ξε,u  (t), ε

(1) k¯ m−2 ηε,uε (t)).



13.2.2.3 Second Type Short Time Ergodic Theorems for Singularly Perturbed Multi-Alternating Regenerative Processes Based on the Second Time Compression Factor. Let us describe the asymptotic behaviour of the probabilities   are Pε,i (tε uε , A) for the case, where the conditions k¯ m−2 T¯ 4 and k¯ m−2 S0 or k¯ m−2 S∞ satisfied and 0 ≤ tε → ∞ as ε → 0 such that tε  → t ∈ (0, ∞) as ε → 0. k¯ m−2 wε

(13.83)

The corresponding limiting stationary probabilities for perturbed multi-alternating regenerative processes have the following form, for γ  ∈ (0, ∞) and β  = 0 or β  = ∞, t ∈ (0, ∞) i ∈ X, and A ∈ BZ ,

358

13 Ergodic theorems for perturbed MARP (0,0) k¯ m−2 π 0,i (t,



A) =

j,r ∈ k¯

m−2

X

(0,0) k¯ m−2 p0,i j k¯ m−2 π 0, jr (t,

A),

(13.84)

where (0,0) k¯ m−2 π 0, jr (t,

A)  k¯ m−2 π0,1 (A)

⎧ ⎪ ⎪ ⎪ ⎨ ⎪

for A ∈ BZ, r 0 for A ∈ BZ, r = −t/e0,2 )  (A) for A ∈ B , r π (1 − e ⎪ Z k¯ m−2 0,1 ⎪ ⎪ ⎪ e−t/e0,2 ¯ π  (A) for A ∈ B Z, r ⎩ k m−2 0,2 and

(∞,∞) (t, k¯ m−2 π 0,i



A) =

j,r ∈ k¯

m−2

X

= = = =

k m−1, , j = k m−1, k m, j = k m−1, k m−1, , j = k m, k m, j = k m,

(∞,∞) k¯ m−2 p0,i j k¯ m−2 π 0, jr (t,

A),

(13.85)

(13.86)

where (∞,∞) k¯ m−2 π 0, jr (t,

A)

 (A) ⎧ e−t/e0,1 k¯ m−2 π0,1 for A ∈ BZ, r ⎪ ⎪ ⎪ ⎨ (1 − e−t/e0,1 ) ¯ π  (A) for A ∈ BZ, r ⎪ k m−2 0,2 = ⎪ 0 for A ∈ BZ, r ⎪ ⎪  (A) ⎪ π for A ∈ BZ, r ¯ ⎩ k m−2 0,2

= = = =

k m−1, j = k m−1, k m, j = k m−1, k m−1, j = k m, k m, j = k m,

(13.87)

 (A), are given by the relations and the limiting stationary probabilities k¯ m−2 π0,r (13.29)–(13.31). As above, the corresponding time compression factor uε = k¯ m−2 uε is given by the relation (13.32). The following theorem, whose proof is similar to the proof of Theorem 13.5, takes place.

Theorem 13.8 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R, and k¯ m−2 M,       ¯ k¯ m−2 T4 , k¯ m−2 Xγ (for some γ = k¯ m−2 γ ∈ (0, ∞)), and k¯ m−2 Sβ (for β = 0 or β = ∞) be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Then, for any i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0 such that tε /wε → t ∈ (0, ∞) as ε → 0: (i) If k¯ m−2 γ  ∈ (0, ∞), k¯ m−2 β  = 0, Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} → (ii) If k¯ m−2 γ  ∈ (0, ∞),

k¯ m−2 β



(0,0) k¯ m−2 π 0,i (t,

= ∞,

A) as ε → 0.

(13.88)

13.2 Ergodic theorems for singularly perturbed MARP

359

Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} →

(∞,∞) (t, k¯ m−2 π 0,i

A) as ε → 0.

(13.89)

Let us also consider the cases, where γ  = 0, β  = 0 or γ  = ∞, β  = ∞, In these cases, the corresponding limiting stationary probabilities for perturbed multi-alternating regenerative processes are given by relations (13.29)–(13.31). Recall also the corresponding time compression factor uε = k¯ m−2 uε given by the relation (13.32). The following theorem, whose proof is similar to the proof of Theorem 13.4, takes place. Theorem 13.9 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R, and       ¯ k¯ m−2 M, k¯ m−2 T4 , and k¯ m−2 Xγ , k¯ m−2 Sβ (for (γ , β ) = (k¯ m−2 γ , k¯ m−2 β ) = (0, 0) or     (γ , β ) = (k¯ m−2 γ , k¯ m−2 β ) = (∞, ∞)) be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Then, for any i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0 such that tε /wε → t ∈ (0, ∞) as ε → 0: (i) If k¯ m−2 γ  = 0, k¯ m−2 β  = 0, Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} → (ii) If k¯ m−2 γ  = ∞,

k¯ m−2 β



(0,0) (A) k¯ m−2 π0

as ε → 0.

(13.90)

= ∞,

Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} →

(∞,∞) (A) k¯ m−2 π0

as ε → 0.

(13.91)

13.2.2.4 Third Type Short Time Ergodic Theorems for Singularly Perturbed Multi-Alternating Regenerative Processes Based on the Second Time Compression Factor. Let us describe the asymptotic behaviour of the probabilities   are Pε,i (tε uε , A) for the case, where the conditions k¯ m−2 T¯ 4 and k¯ m−2 S0 or k¯ m−2 S∞ satisfied, and 0 ≤ tε → ∞ as ε → 0 such that tε  → 0 as ε → 0. k¯ m−2 wε

(13.92)

The corresponding limiting stationary probabilities for perturbed multi-alternating regenerative processes are given by the relations (13.66) and (13.67). The corresponding time compression factor uε = k¯ m−2 uε is given by the relation (13.32). The following theorem, whose proof is analogous to the proof of Theorem 13.2, takes place.

360

13 Ergodic theorems for perturbed MARP

Theorem 13.10 Let conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R, and k¯ m−2 M,      ¯ k¯ m−2 T4 , k¯ m−2 Xγ (for some γ = k¯ m−2 γ ∈ (0, ∞)), and k¯ m−2 Sβ (for β = 0 or  β = ∞) hold for multi-alternating regenerative processes (ξε (t), ηε (t)). Then, for any i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0 such that tε /wε → 0 as ε → 0, Pε,i (tε k¯ m−2 uε , A) = Pi {ξε (tε k¯ m−2 uε ) ∈ A} →

 k¯ m−2 π¯ 0,i (A)

as ε → 0.

(13.93)

13.2.2.5 Ergodic Theorems for Super-Singularly Perturbed Multi-Alternating Regenerative Processes. It should be noted that the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , imply that the conditions k¯ n G1 and k¯ n H1 are satisfied, for every n = 0, . . . , m − 1. Thus, the phase space k¯ n X is, for every ε ∈ (0, 1], one class of communicative states for the embedded semi-Markov process k¯ n ηε (t). This makes it impossible to obtain ergodic theorems for super-singularly perturbed multi-alternating regenerative processes similar to ergodic Theorems 9.1–9.8 for super-singularly perturbed alternating processes. We hope to present such theorems in future papers.

13.3 Ergodic Theorems for Perturbed Multi-Alternating Regenerative Processes Based on Embedded Regenerative Processes In this section, we present ergodic theorems for perturbed multi-alternating regenerative processes based on perturbed embedded regenerative processes. We also clarify the relationship between ergodic theorems for perturbed multi-alternating regenerative processes based on embedded alternating regenerative processes and embedded regenerative processes.

13.3.1 Ergodic Theorems Based on Perturbed Embedded Regenerative Processes 13.3.1.1 Ergodic Theorems Based on the Use of Embedded Regenerative Processes with Totally Removed Virtual Transitions. In this subsection, we use the (1) (1) embedded regenerative processes (k¯ m−1 ξε, uˆ ε (t), k¯ m−1 ηε, uˆ ε (t)). The corresponding limiting stationary probabilities k¯ m−1 π0 (A) for the perturbed multi-alternating regenerative processes have the following form: ∫ ∞ 1 π (A) = (13.94) k¯ m−1, l q0,k m (s, A)m(ds), for A ∈ BZ, k¯ m−1 0 k¯ m−1 f0,k m 0

13.3 Ergodic theorems for perturbed MARP based on embedded RP

361

where the expectation k¯ m−1, l f0,r and the function k¯ m−1, l q0,r (s, A) are given by the relations (12.172) and (12.174), respectively. In this case, the corresponding time compression factor uˆε is given by the relations (12.145) and (12.157), that is, uˆε =

k¯ m−1 uε,k m

=

m−2 

−1 k¯r p¯ε,k m k m

uε,km .

(13.95)

r=0

The following theorem takes place. Theorem 13.11 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R, and k¯ m−1 M hold for multi-alternating regenerative processes (ξε (t), ηε (t)). Then, for any i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0, Pε,i (tε k¯ m−1 uε,km , A) = Pi {ξε (tε k¯ m−1 uε,km ) ∈ A} →

k¯ m−1 π0 (A)

as ε → 0.

(13.96)

Proof From Lemmas 12.11 and 12.12 it follows that in this case Theorem 2.3 can be applied directly to the embedded regenerative processes with transition period (k¯ m−1 ξε, uˆ ε (t), k¯ m−1 ηε, uˆ ε (t)). The corresponding stopping probabilities appearing in Theorem 2.3 are in this case equal to 0 for ε ∈ (0, 1].  13.3.1.2 Ergodic Theorems Based on the Use of Embedded Regenerative Processes with Partially Removed Virtual Transitions. In this subsection, we use (1) the embedded regenerative processes (k¯ m−1 ξε,(1) uˆ ε (t), k¯ m−1 ηε, uˆ ε (t)). The corresponding limiting stationary probabilities k¯ m−1 π0 (A) for perturbed multialternating regenerative processes have the following form: ∫ ∞ 1   (13.97) k¯ m−1 q0,k m (s, A)m(ds), for A ∈ BZ, k¯ m−1 π0 (A) =  k¯ m−1 f0,k m 0   and the function k¯ m−1 q0,k (s, A) are given by the where the expectation k¯ m−1 f0,k m m relations (12.216) and (12.218), respectively. In this case, the corresponding time compression factor uˆε is given by the relations (12.188) and (12.201), that is,

uˆε =

 k¯ m−1 uε,k m

=

k¯ m−1 uε,k m

(13.98)

The following theorem takes place. Theorem 13.12 Let the conditions G1 , H1 , I1 , IH,1 , L1 , LH,1 , J1 , K1 , R, and k¯ m−1 M be satisfied for the multi-alternating regenerative processes (ξε (t), ηε (t)). Then, for any i ∈ X, A ∈ Γ and 0 ≤ tε → ∞ as ε → 0,

362

13 Ergodic theorems for perturbed MARP   Pε,i (tε k¯ m−1 uε,k , A) = Pi {ξε (tε k¯ m−1 uε,k ) ∈ A} m m

→

 k¯ m−1 π0 (A)

as ε → 0.

(13.99)

Proof From Lemmas 12.15 and 12.16 it follows that in this case Theorem 2.3 can be applied directly to the embedded regenerative processes with transition period  (k¯ m−1 ξε, uˆ  (t), k¯ m−1 ηε, (t)). uˆ ε ε The corresponding stopping probabilities appearing in Theorem 2.3 are in this case equal to 0 for ε ∈ (0, 1]. 

13.3.2 Relationship Between Ergodic Theorems Based on Perturbed Embedded Alternating Regenerative Processes and Embedded Regenerative Processes 13.3.2.1 Relationship Between Theorems 13.11 and 13.12. The condition k¯ m−1 M is assumed to hold in both Theorems 13.11 and 13.12. It is obvious that the set k¯ m−2 K[k m−1 ]

= {k m−1, k m }.

(13.100)

According to Lemma 11.1, the relation (13.100) implies that the normalisation  functions uˆε = k¯ m−1 uε,km and uˆε = k¯ m−1 uε,k coincide, i.e., m uˆε = uˆε , ε ∈ (0, 1].

(13.101)

The relation (13.101) implies that the ergodic relations (13.96) and (13.99) describe the asymptotic behaviour of the same probabilities Pi {ξε (tε uˆε ) ∈ A} = Pi {ξε (tε uˆ ε ) ∈ A}, as ε ∈ (0, 1], for i ∈ X, A ∈ Γ. Thus, the ergodic relations given in Theorems 13.11 and 13.12 imply that the following relation holds for the corresponding limiting stationary distributions, k¯ m−1 π0 (A)

=

 k¯ m−1 π0 (A),

for A ∈ Γ.

(13.102)

In fact, Theorems 13.11 and 13.12 are equivalent. The only difference is in recurrent algorithms used to calculate stationary distributions k¯ m−1 π0 (A) and k¯ m−1 π0 (A). In Theorem 13.11, the corresponding algorithm of phase space reduction for modulating semi-Markov processes is based on the total removal of virtual transitions, whereas in Theorem 13.12, the corresponding algorithm of phase space reduction for modulating semi-Markov processes is based on partial removal of virtual transitions. According to the relation (13.102), both algorithms give the same limiting stationary probabilities, for A ∈ Γ. 13.3.2.2 Relationship Between Theorems 13.1, 13.2 and Ergodic Theorems 13.11, 13.12, for Regularly Perturbed Multi-Alternating Regenerative Processes. First, let us clarify relationship between Theorems 13.2 and 13.12.

13.3 Ergodic theorems for perturbed MARP based on embedded RP

363

In this case, we assume that all conditions of these theorems are satisfied. According to the condition k¯ m−2 H, the phase space k¯ m−2 X = {k m−1, k m } is one class of communicative states. This implies that set k¯ m−2 K[k m−1 ]

=

k¯ m−2 X.

(13.103)

In Theorem 13.2, the condition k¯ m−2 M is assumed to be satisfied. In Theorem 13.12, the condition k¯ m−1 M is assumed to be satisfied, i.e., it is assumed that the condition k¯ m−2 M is satisfied and, in addition, the following relation holds:  k¯ m−1 u˜ ε,k m−1  k¯ m−1 u˜ ε,k m

=

k¯ m−2 u˜ ε,k m−1 k¯ m−2 u˜ ε,k m

→

k¯ m−2 w0,k m−1,k m

∈ [0, ∞) as ε → 0,

where, according to the relations (11.20) and (13.103), for i ∈  k¯ m−1 u˜ ε,i

=

k¯ m−2 K[k m−1 ]

−1  k¯ m−2 p¯ε,ii k¯ m−2 uε,i .

(13.104) =

k¯ m−2 X,

(13.105)

Note that the time compression factors uε and uˆε take the following forms:

and

uε =

 k¯ m−2 uε,k m−1

uˆε =

 k¯ m−1 uε,k m

+

=

 k¯ m−2 uε,k m ,

(13.106)

 k¯ m−1 u˜ ε,k m .

(13.107)

The relation (13.105) implies that the following relation holds:  k¯ m−1 u˜ ε,k m  k¯ m−1 u˜ ε,k m−1

=

 −1 k¯ m−2 uε,k m k¯ m−2 p¯ε,k m k m  −1 k¯ m−2 uε,k m−1 k¯ m−2 p¯ε,k m−1 k m−1

.

(13.108)

The relations (13.104), (13.105) and the conditions k¯ m−2 Xγ  , k¯ m−2 Sβ  imply that the following relation holds, for (k¯ m−2 γ , k¯ m−2 β )  (0, ∞), (∞, 0), −1 k¯ m−2 w0,k m,k m−1

= = =

 k¯ m−1 u˜ ε,k m lim ε→0 k¯ u˜  m−1 ε,k m−1  −1 k¯ m−2 uε,k m k¯ m−2 p¯ε,k m k m lim lim −1 ε→0 k¯ ε→0 ¯ u k m−2 p¯ε,k m−1 k m−1 m−2 ε,k m−1   k¯ m−2 γ k¯ m−2 β ∈ (0, ∞].

(13.109)

Note also that all limits in relation (13.109) exist and can be calculated using Lemmas 8.21 –8.91 . Relation (13.109) implies that, in this case, k¯ m−2 γ



,

k¯ m−2 β



∈ (0, ∞].

(13.110)

364

13 Ergodic theorems for perturbed MARP

It is assumed in Theorem 13.2 that the condition k¯ m−2 T4 is satisfied, i.e.,   k¯ m−2 p¯0,k m−1 k m−1 + k¯ m−2 p¯0,k m k m > 0. This relation and the relation (13.110) imply  > 0. that the probability k¯ m−2 p¯0,k m−1 k m−1  > 0 and, thus, the parameter k¯ m−2 β  ∈ It is possible that the probability k¯ m−2 p¯0,k m km (0, ∞) (the case of regularly perturbed modulating semi-Markov processes k¯ m−2 ηε (t)),  = 0 and, thus, k¯ m−2 β  = ∞ (the case of semi-regularly perturbed or k¯ m−2 p¯0,k m km modulating semi-Markov processes k¯ m−2 ηε (t)). Recall the time compression factor,  k¯ m−2 vε

=

−1 k¯ m−2 p¯ε,k m−1 k m−1

+

−1 k¯ m−2 p¯0,k m k m

+

−1 k¯ m−2 p¯ε,k m k m .

(13.111)

In this case,  k¯ m−2 vε

→ 

=

 k¯ m−2 v0

−1 k¯ m−2 p¯0,k m−1 k m−1

∞

∈ (0, ∞) if if

k¯ m−2 β



∈ (0, ∞),  = ∞. β k¯ m−2

(13.112)

The relations (13.105), (13.110) and the above remarks imply that, in this case,  cε,+

= =

u vε ε uˆε

→

=

 u + k¯ m−2 uε,k ¯ m  k m−2 ε,k m−1 k¯ m−2 vε  u k¯ m−1 ε,k

 k¯ m−2 p¯ε,k m k m 1+ 1+  k¯ m−2 p¯ε,k m−1 k m−1  c0,+ = (1 + k¯ m−2 β −1 )(1 +

m

 k¯ m−2 uε,k m−1  k¯ m−2 uε,k m −1 ) ∈ [1, ∞) k¯ m−2 γ

as ε → 0.

(13.113)

Let us choose tε, tˆε ≥ 0 connected by the following equality, for ε ∈ (0, 1],  −1 tˆε = tε vε−1 cε,+ or, equivalently, tε = tˆε vε cε,+ .

(13.114)

In this case, for ε ∈ (0, 1],  tˆε uˆε = tε vε−1 cε,+

vε uε = tε uε ,  cε,+

(13.115)

and, thus, for ε ∈ (0, 1] and i ∈ X, A ∈ Γ, Pi {ξε (tε uε ) ∈ A} = Pi {ξε (tˆε uˆ ε ) ∈ A}.

(13.116)

If tˆε → ∞ as ε → 0, then tε → ∞ as ε → 0. Thus, the ergodic relations given in Theorems 13.2 and 13.12 and the relation (13.116) imply that (γ,β ) (A) for A ∈ Γ, γ  ∈ (0, ∞), β  ∈ (0, ∞],  k¯ m−2 π0 (13.117) ¯k m−1 π0 (A) = (∞,∞) (A) for A ∈ Γ, γ  = ∞, β  ∈ (0, ∞]. k¯ m−2 π0

13.3 Ergodic theorems for perturbed MARP based on embedded RP

365

If β  ∈ (0, ∞), the relations, tˆε → ∞ as ε → 0 and tε → ∞ as ε → 0, are equivalent. Thus, the corresponding ergodic relations given in Theorems 13.2 and 13.12 are equivalent, if γ  ∈ (0, ∞], β  ∈ (0, ∞). If β  = ∞, the relation tˆε → ∞ as ε → 0 implies that tε → ∞ as ε → 0, while the relation tε → ∞ as ε → 0 implies that tˆε → ∞ as ε → 0 only if tε vε−1 → ∞ as ε → 0. In this case, the corresponding ergodic relations given in Theorems 13.2 are stronger than the ergodic relation given in Theorem 13.12, if γ  ∈ (0, ∞], β  = ∞. Indeed, in the case where tε → ∞ as ε → 0 and tε vε−1 → t ∈ [0, ∞) as ε → 0, the corresponding ergodic relations given in Theorems 13.2 still take place. But,  ∈ [0, ∞) as ε → 0, and, thus, Theorem 13.12 does not work in this case. tˆε → tc0,+ It should also be noted that Theorem 13.2 does not work in the cases where γ  = ∞, β  = 0 or γ  = 0, β  = ∞, while Theorem 13.12 works. There is also a difference in the implementations of the recurrent algorithm used for computing the stationary distributions in these theorems. In fact, Theorem 13.12 requires an additional step of phase space reduction and the corresponding calculation of limiting characteristics under conditions like J, K, and R. So, the situation is ambiguous. In the main case where γ , β  ∈ (0, ∞), Theorems 13.2 and 13.12 are equivalent. However, there are cases where either Theorem 13.2 or Theorem 13.12 works better. It should also be taken into account that the condition k¯ m−2 M is used in Theorem 13.2. In this case, the states k m−1 and k m are not ordered by their absorbing rates. Theorem 13.2 also covers the case where the state k m−1 is the most absorbing state, i.e., condition k¯ m−2 M is satisfied and, in addition, the following relation holds: k¯ m−2 u˜ ε,k m k¯ m−2 u˜ ε,k m−1

=

k¯ m−2 uε,k m k¯ m−1 uε,k m

→

k¯ m−2 w0,k m,k m−1

∈ [0, ∞) as ε → 0.

(13.118)

Theorems 13.2 and 13.12 can be reformulated for this case by formally swapping symbols k m−1 and k m in the corresponding relations and formulas. Similarly, the comparative analysis presented above can be implemented for the above case by formally swapping symbols k m−1 and k m in the relations (13.100)–(13.117). A comparative analysis, similar to that presented above for Theorems 13.2 and 13.12, can be implemented for Theorems 13.1 and 13.11. Taking into account the results of the comparative analysis of Theorems 13.11 or 13.12 presented in Sect. 13.3.2.1, we can conclude that all four Theorems 13.1, 13.2, 13.11, and 13.12 are equivalent in the basis case where γ , β  ∈ (0, ∞) and γ, β ∈ (0, ∞) (in fact, it is possible to show that these relations are true or not true at the same time).

366

13 Ergodic theorems for perturbed MARP

13.3.2.3 Relationship Between Theorem 13.12 and the Super-Long Time Ergodic Theorem 13.3 for Singularly Perturbed Multi-Alternating Regenerative Processes. The comparative analysis implemented in Sect. 13.3.2.2 can also be implemented for Theorems 13.12 and 13.3. In this case, the conditions of Theorem 13.3 include the assumption that tε vε−1 → ∞ as ε → 0. Therefore, the corresponding ergodic relations given in Theorems 13.3 and 13.12 are equivalent, if γ  ∈ (0, ∞], β  ∈ (0, ∞]. As above, it should also be noted that Theorem 13.3 does not work in the cases, where γ  = ∞, β  = 0 and γ  = 0, β  = ∞, while Theorem 13.12 works. There is also a difference in the implementations of the recurrent algorithm used for computing the stationary distributions in these theorems. In fact, Theorem 13.12 requires an additional step of phase space reduction and the corresponding calculation of limiting characteristics under conditions like J, K, and R. As above, Theorems 13.3 and 13.12 can be reformulated for the case, where the state k m−1 is the most absorbing state (i.e., the relation (13.118) holds) by formally swapping symbols k m−1 and k m in the corresponding relations and formulas. Similarly, the comparative analysis presented in the present above can be implemented for the above case by formally swapping symbols k m−1 and k m in the relations (13.100)–(13.117). 13.3.2.4 Relationship Between Theorem 13.12 and the Long and Short Time Ergodic Theorems 13.4–13.10 for Singularly Perturbed Multi-Alternating Regenerative Processes. The comparative analysis implemented in Sect. 13.3.2.2 can also be implemented for Theorems 13.12 and 13.4–13.10. In this case, the situation is fundamentally different. The conditions of Theorems 13.4–13.10 include the assumption that tε vε−1 → t ∈ [0, ∞) as ε → 0. In this case, Theorems 13.4–13.10 work, but Theorem 13.12 does not work. In this case, the use of embedded alternating regenerative processes gives a number of new long and short time ergodic theorems for singularly perturbed multialternating regenerative processes, which cannot be obtained using embedded regenerative processes.

Appendix A

Perturbed Renewal Equation

In this appendix, we present some results related to generalisation of the classical renewal theorem to a model of a perturbed renewal equation. The renewal theorem for the perturbed renewal equation is one of the main tools used to obtain ergodic theorems for perturbed alternating and multi-alternating regenerative processes, which are the main subjects of this book.

A.1 Renewal Equation and Renewal Theorem In this section, we present some key results of renewal theory, in particular the renewal theorem. Its final form was given by Feller (1971).

A.1.1 Renewal Equation A.1.1.1 Renewal Equation. We denote by L the class of real-valued measurable (Borel) functions defined on [0, ∞) and bounded on any finite interval. The following equation, known as the renewal equation, plays a fundamental role in renewal theory and its applications, ∫ t x(t) = q(t) + x(t − s)F(ds), t ≥ 0, (A.1) 0

where: (a) F(x) is a distribution function on [0, ∞), which is not concentrated at zero, (b) q(t) is a function from the class L, and (c) the solution of the renewal equation (A.1) is sought in the class L. Denote by F (∗n) (t) the n-fold convolution of the distribution function F(x). It is the distribution function defined by the following recurrent relation:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0

367

368

A Perturbed Renewal Equation

F (∗n) (t) =



I(t ≥ 0), t ≥ 0, for n = 0, F (∗(n−1)) (t) ∗ F(t), t ≥ 0, for n = 1, 2, . . . ,

(A.2)

where the convolution F(t) = F1 (t) ∗ F2 (t) of two proper or improper distribution functions F1 (t) and F2 (t) concentrated on the non-negative half-line is defined by the following relation: ∫ t F (t − s)F2 (ds) for t ≥ 0, F(t) = 0 1 0 for t < 0. The following function, called the renewal function, plays an important role in the theory: ∞

U(t) = F (∗r) (t), t ≥ 0. (A.3) r=0

It can easily be shown that the series in (A.3) converges for all t ≥ 0 and, moreover, U(t) ≤ U1 + U2 t for some finite constants U1 and U2 . Thus, function U(t) has a linear rate of growth in t. Also note that by the definition U(t) is a non-decreasing right-continuous function. Let us also define the renewal measure U(A) on the Borel σ-algebra B+ of the interval [0, ∞) generated by the renewal function in the usual way. This measure is defined on intervals as U((a, b]) = U(b) − U(a), 0 ≤ a ≤ b < ∞. Note that this measure always has an atom in zero since series (A.3) includes the 0-fold convolution F (∗0) (t). This convolution is by definition a distribution function with a unit jump at zero. As is well known, there exists the unique solution of the renewal equation in the class L, which has the following form: ∫ t x(t) = q(t − s)U(ds), t ≥ 0. (A.4) 0

A.1.2 Non-Arithmetic Distribution Functions A.1.2.1 Definitions. Let Uh,d be the grid of points on real line with step h > 0 and shift d ∈ [0, h), i.e., Uh,d = {uh,d,n = nh − d, n = 0, ±1, . . .}.

(A.5)

Let also F be the family of distribution functions F(·) on the real line not concentrated at zero, that is, such that F(0) − F(0 − 0) < 1. Denote ∞

(F(nh − d) − F(nh − d − 0)). (A.6) F(Uh,d ) = n=−∞

A.1 Renewal equation and renewal theorem

369

Let us introduce the set, A0 = {h > 0 : F(Uh,0 ) = 1}.

(A.7)

Definition A.1 The distribution function F(·) ∈ F is strictly arithmetic if the set A0  ∅ and weakly non-arithmetic if the set A0 = ∅. If the distribution F(·) ∈ F is strictly arithmetic, then h0,F = max(h : h ∈ A0 ) < ∞.

(A.8)

The parameter h0,F is called the span of the strictly arithmetic distribution function F(·). Let us introduce set, A = {h > 0 : F(Uh,d )) = 1 for some d ∈ [0, h)}.

(A.9)

Note that for a given h > 0, either F(Uh,d )) < 1 for all d ∈ [0, h) or there is unique d = dh ∈ [0, h) such that F(Uh,d )) = 1. Definition A.2 The distribution function F(·) ∈ F is arithmetic if the set A  ∅ and non-arithmetic if the set A = ∅. If the distribution F(·) ∈ F is arithmetic, then hF = max(h : h ∈ A) < ∞.

(A.10)

The parameter hF is called the span of the arithmetic distribution function F(·), and the parameter dF = dh F , such that F(Uh F ,dF ) = 1, is called the shift of the arithmetic distribution function F(·). Obviously, set A0 ⊆ A and, thus, any strictly arithmetic distribution function F(·) is arithmetic. In this case, span h0,F ≤ hF . A.1.2.2 Criteria of Non-arithmeticity. Let ϕ(z) be the characteristic function of the distribution function F(u), i.e., ∫ ϕ(z) = eizu F(du), z ∈ R1 . (A.11) R1

The following lemma (see, for example, Feller (1971)) presents necessary and sufficient criteria for the distribution function F(u) ∈ F to be weakly non-arithmetic or non-arithmetic. Lemma A.1 Let a distribution function F(·) ∈ F has the characteristic function ϕ(z). Then: (i) The distribution function F(u) is weakly non-arithmetic if and only if ϕ(z)  1, z  0. Otherwise, if the distribution function F(u) is strictly arithmetic, then ϕ(z) is a periodic function with the period 2π h , where h = h0,F is the span for this 2π distribution function, ϕ( h ) = 1, and ϕ(z)  1 for 0 < z < 2π h .

370

A Perturbed Renewal Equation

(ii) The distribution function F(u) is non-arithmetic if and only if |ϕ(z)| < 1, z  0. Otherwise, if the distribution function F(u) is arithmetic, then ϕ(z)eizdh is a periodic function with the period 2π h , where h = hF and dh = dh F are, respectively, the span 2π i 2π h dh = 1, and, thus, |ϕ( and the shift for this distribution function, ϕ( 2π h )e h )| = 1, 2π while |ϕ(z)| < 1 for 0 < z < h . A1.2.3 Examples. According to the Lebesgue decomposition, any distribution function F(x), x ∈ R1 can be represented in the form F(x) = qa Fa (x) + qd Fd (x) + qs Fs (x),

(A.12)

where: (a) Fa (x) is an absolutely continuous distribution function, which can be ∫x represented in the form of Lebesgue integral, i.e., Fa (x) = −∞ f (y)m(dy), x ∈ R1 , where f (y) is a non-negative, Borel measurable function and m(dy) is the Lebesgue measure on the Borel σ-algebra of real line BR1 , (b) Fd (x) is a discrete distribution  function, i.e., Fd (x) = an ≤x pk , x ∈ R1 , where an ∈ R1, n ∈ N and pn ≥ 0, n ∈ N,  n∈N pn = 1, N is a finite or countable set of natural numbers, (c) Fs (x) is a singular continuous distribution function such that the corresponding measure F(A) generated by this distribution function on the σ-algebra BR1 is concentrated on some zero-set S ∈ BR1 , that is, F(S) = 1 and m(S) = 0, (d) qa, qd, qs ≥ 0, qa + qd + qs = 1. Any distribution function, for which qa + qs > 0, is non-arithmetic. Any distribution function F(x) = I(x ≥ a) concentrated in point a  0 is strictly arithmetic, with span h0,F = |a|. Let F(x) = p1 I(x ≥ a1 ) + p2 I(x ≥ a2 ), where p1, p2 > 0, p1 + p2 = 1, a1 < a2 , be a distribution function concentrated in two points. Any such distribution function is arithmetic, with span hF = a2 − a1 and shift dF = − max(nh : nh ≤ 0). Such distribution function is weakly arithmetic if a1 = 0 or a2 = 0, with span h0,F = a2 − a1 . Also, such distribution function is weakly arithmetic, if a1, a2  0 and a1 /a2 = r1 /r2 is a rational number (represented in the irreducible form), with span h = |a1 /r1 | = |a2 /r2 |. However, if a1, a2  0 and a1 /a2 = a is an irrational number, then the distribution function F(x) is arithmetic but not strictly arithmetic, i.e., it is weakly non-arithmetic.

A.1.3 Renewal Theorem An important role in probability theory is played by the classical renewal theorem, which describes the asymptotic behaviour of the solution x(t) of the equation (A.1) for t → ∞. This theorem has a long history of development. The final form and the proof of the renewal theorem were given by Feller (1971). Let q(t) be a function from the class L. Define upper and lower Riemann sums, for h > 0,

A.1 Renewal equation and renewal theorem

Σh+ (q(·)) =

∞

sup

r=0 rh ≤t ≤(r+1)h

371

q(t), Σh− (q(·)) =

∞

r=0

inf

rh ≤t ≤(r+1)h

q(t).

(A.13)

Definition A.3 A function q(t) from the class L is directly Riemann integrable on the interval [0, ∞), if the series (A.13) converge for all sufficiently small h and Σh+ (q(·)) − Σh− (q(·)) → 0 as h → 0. If function q(t) is directly Riemann integrable on interval [0, ∞), then there exists a finite limit, ∫ lim Σh± (q(·)) =

h→0

∞

q(t)dt.

(A.14)

0

A real-valued function q(t) defined on the interval [0, ∞) is directly Riemann integrable if and only if the following condition is satisfied:  Y1 : (a) q(·) ∈ L, (b) limT →∞ h r ≥T /h suprh ≤t ≤(r+1)h |q(t)| = 0, for some h > 0, (c) q(·) is continuous on almost everywhere with respect to the Lebesgue measure on the Borel σ-algebra B+ of the interval [0, ∞). If the function q(t) is directly Riemann integrable on [0, ∞), then it is also absolutely integrable on [0, ∞) in the usual way, i.e.: (a) it is Riemann integrable on ∫ T  any finite interval [T , T ], 0 ≤ T  ≤ T  < ∞, (b) T  |q(s)|ds → 0 as 0 ≤ ∫T ∫∞ T , T  → ∞, and, thus, there exists a finite limit, limT →∞ 0 q(t)dt = 0 q(t)dt. If q(t) is a non-negative non-increasing function, then it is directly Riemann integrable on [0, ∞) if and only if it is absolutely integrable on [0, ∞) in the usual way. Also, if q(t) is a function continuous almost everywhere with respect to the Lebesgue measure on [0, ∞) and |q(t)| ≤ q+ (t), t ≥ 0, where the function q+ (t), t ≥ 0 is a non-negative non-increasing function absolutely integrable on [0, ∞), then the function q(t) is directly Riemann integrable on [0, ∞). This follows from the following relation: ∫ ∞

sup |q(s)| ≤ lim q+ (s)ds = 0. (A.15) lim h T →∞

T →∞ T −h

r ≥T /h rh ≤s ≤(r+1)h

Let us introduce the first moment of the distribution function F(·), ∫ ∞ eF = tF(dt). 0

The conditions of the renewal theorem are as follows: Y2 : F(x) is a weakly non-arithmetic distribution function. Y3 : eF < ∞, and Y4 : q(t) is a directly Riemann integrable function on [0, ∞).

(A.16)

372

A Perturbed Renewal Equation

Now we are prepared to formulate the classical variant of the renewal theorem in the final form given by Feller (1971). Theorem A.1 Let the conditions Y2 –Y4 be satisfied. Then, ∫ ∞ 1 q(s)ds as t → ∞. x(t) → x(∞) = eF 0

(A.17)

It is also worth to mention the case that is important for applications of the renewal theorem to ergodic theorems for regenerative-type processes. We say that non-negative function q(·) is majorised by the tail probability function 1 − F(·) on a set S ⊆ [0, ∞) if q(t) ≤ 1 − F(t), for t ∈ S. If S = [0, ∞), then the condition Y4 is satisfied if the function q(·) is continuous almost∫ everywhere with respect to the Lebesgue measure on B+ and the first moment ∞ eF = 0 (1 − F(t))dt < ∞.

A.2 Perturbed Renewal Equation and Renewal Theorem In this section we formulate theorems generalising the classical renewal theorem to the model of perturbed renewal equation. This generalisation was given in Silvestrov (1976, 1978, 1979). The detailed presentation of related results and the proofs can be found in these papers and the book Gyllenberg and Silvestrov (2008).

A.2.1 Perturbed Renewal Equation Consider for each ε ∈ (0, 1] the renewal equation, ∫ t xε (t) = qε (t) + xε (t − s)Fε (ds), t ≥ 0,

(A.18)

0

where: (a) qε (t) is a real-valued function on [0, ∞) from the class L, (b) Fε (s) is a proper or improper distribution function on [0, ∞) that is not concentrated in 0, i.e., Fε (∞) ≤ 1 and Fε (0) < 1. The general facts and definitions concerning the renewal equation carry over without any changes to the case of an improper renewal equation generated by an improper distribution function. The renewal function Uε (t) generated by the distribution function Fε (s) can be defined using the following variant of the relation (A.3): Uε (t) =

∞

n=0

Fε(∗n) (t), t ≥ 0.

(A.19)

A.2 Perturbed Renewal Equation and Renewal Theorem

373

The renewal equation (A.18) has a unique solution xε (t) in the class L, which is given by the following version of the relation (A.4): ∫ t xε (t) = qε (t − s)Uε (ds), t ≥ 0. (A.20) 0

A.2.2 Renewal Theorem for Perturbed Renewal Equation As usual, the symbol Fε (·) ⇒ F0 (·) as ε → 0 means weak convergence of distribution functions (proper or improper), that is, point-wise convergence at each point of continuity of the limiting distribution function. In what follows, we use the symbol ε → 0 as a short version of symbol 0 < ε → 0. Let us define the first moments of the distribution function Fε (s), for ε ∈ (0, 1], ∫ ∞ eε,F = sFε (ds). (A.21) 0

We assume that the following conditions are satisfied: Y5 : Fε (·) ⇒ F0 (·) as ε → 0, where F0 (s) is a proper, weakly non-arithmetic distribution function. ∫∞ Y6 : (a) eε,F < ∞, for ε ∈ (0, 1], (b) eε,F → e0,F = 0 sF0 (ds) < ∞ as ε → 0. Y7 : (a ) limε→0 sup0≤s ≤T |qε (s)| < ∞, for T ≥ 0,  (b) limT →∞ limε→0 h r ≥T /h suprh ≤s ≤(r+1)h |qε (s)| = 0, for some h > 0. us

Y8 : qε (·) −→ q0 (·) as ε → 0, for points s ∈ U[q · (·)], where: (a) q0 (t), t ∈ R+ is a real-valued function from the class L, (b) U[q · (·)] is a Borel subset of R+ , such that the Lebesgue measure m(U¯ [q · (·)]) = 0, (c) the function q0 (t), t ∈ R+ , is directly Riemann integrable on [0, ∞). It is worth noting that the notation U[q · (·)] is used to show this set of convergence is actually determined by the family of functions qε (·), ε ∈ (0, 1]. us The definition of locally uniform convergence denoted by the symbol −→ is given in Appendix B. It is also useful to note that the sub-conditions (a) and (b) in the condition Y8 are “independent.” Let, for example, the functions qε (·) ≡ 0(·) for ε ∈ (0, 1], where 0(s) = 0, s ≥ 0, and the function q0 (·) ≡ D(·), where D(s), s ≥ 0 is the Dirichlet function, which us takes the value 1 for rational s and the value 0 for irrational s. Then, qε (·) −→ q0 (·) as ε → 0, for s ∈ U+ , where U+ is the subset of all irrational numbers of R+ and, thus, U¯ + is the subset of rational numbers of R+ . As known, m(U¯ + ) = 0, while the function q0 (·) is discontinuous at all points s ∈ R+ .

374

A Perturbed Renewal Equation

Let, now, the functions qε (·) ≡ D(·) for ε ∈ (0, 1] and q0 (·) ≡ 0(·). In this case, the function q0 (·) is continuous on R+ and qε (s) → q0 (s) as ε → 0, for S ∈ U+ , i.e., the functions qε (·) converge point-wise to the function q0 (·) as ε → 0 at any point s ∈ U+ . However, the functions qε (·) do not converge locally uniformly to the function q0 (·) as ε → 0 at any point s ∈ R+ . According to the condition Y8 , the function q0 (·) belongs to the class L. Also, according to condition Y5 , F0 (·) is a proper distribution function not concentrated in 0. Thus, the following standard renewal equation can be written down ∫ t x0 (t) = q0 (t) + x0 (t − s)F0 (ds), t ≥ 0. (A.22) 0

The conditions Y5 , Y6 , and Y8 are some convergence conditions for the distribution functions Fε (s), their first moments eε,F , and the functions qε (t), respectively, to the corresponding limiting distribution function F0 (s), its first moment e0,F , and the function q0 (t), as ε → 0. This let us consider the equation (A.18) for ε ∈ (0, 1] as a perturbed version of the equation (A.22) and interpret the parameter ε as a perturbation parameter. Let us make some useful remarks concerning the conditions Y5 –Y8 . Remark A.1 The condition Y7 implies that the functions qε (t) are Lebesgue integrable on the interval [0, ∞) for all ε small enough. However, there is no guarantee that these functions are directly Riemann integrable on [0, ∞), for ε ∈ (0, 1]. As a matter of fact, there is no guarantee that these functions are continuous almost everywhere with respect to the Lebesgue measure on the σ-algebra B+ . As far as the limiting function q0 (t) is concerned, this function is not only Lebesgue integrable, but also is directly Riemann integrable on [0, ∞), according to the condition Y8 . Remark A.2 The conditions Y5 –Y8 reduce to the conditions Y2 –Y4 of the classical renewal Theorem A.1, if Fε (s) ≡ F0 (s), s ≥ 0 and qε (t) ≡ q0 (s), s ≥ 0 do not depend on the perturbation parameter ε. The condition Y5 obviously implies that the following relation holds: qε = 1 − Fε (∞) → q0 = 1 − F0 (∞) = 0 as ε → 0.

(A.23)

The following theorem, given in Silvestrov (1976, 1978, 1979), generalises the classical renewal Theorem A.1 to the model of perturbed renewal equation. Theorem A.2 Let the conditions Y5 –Y8 be satisfied. Then, for any 0 ≤ tε → ∞ as ε → 0 such that qε tε → t ∈ [0, ∞] as ε → 0, ∫∞ q0 (s)ds −t/e0, F · 0 as ε → 0. (A.24) xε (tε ) → x0 (∞) = e e0,F Theorem A.2 generalises the Theorem A.1. A new exponential factor appears on the right hand side of the asymptotic relation (A.24). This relation describes

A.2 Perturbed Renewal Equation and Renewal Theorem

375

the asymptotical behaviour of the solution of the renewal equation xε (tε ) as time tε tends to infinity and the perturbation parameter ε → 0. The appearance of the factor e−t/e0, F is caused by a possible deficiency of the distribution function Fε (s) generating the renewal equation (A.18). At the same time, Theorem A.2 imposes some restriction on the rate of growth of time tε , while Theorem A.1 does not impose restrictions on the speed at which tε tends to infinity. The equations (A.18) become proper renewal equations, if qε ≡ 0. In this case, the relation, qε tε → t as ε → 0, automatically holds. Moreover, since qε tε ≡ 0, the parameter t = 0. Note also that in this case tε can tend to infinity in an arbitrary way. The exponential factor on the right hand side of the relation (A.24) takes the value 1. Theorem A.2 reduces to the triangular array generalisation of Theorem A.1. Moreover, if Fε (s) ≡ F0 (s), s ≥ 0 and qε (t) ≡ q0 (t), t ≥ 0 do not depend on the perturbation parameter ε, then Theorem A.2 reduces to Theorem A.1. It is also worth mentioning a case that is important for applications of the above renewal theorem to ergodic theorems for perturbed regenerative processes. This is the case where the following conditions replace, respectively, the conditions Y7 and Y8 : Y9 : |qε (t)| ≤ qε,+ (t), t ≥ 0, for ε ∈ (0, 1], where qε,+ (t), t ≥ 0 is a non-negative and non-increasing ∫ ∞ function, such that: (a1) limε→0 qε,+ (0) < ∞ and (a2) limT →∞ limε→0 T qε,+ (s)ds = 0. us

Y10 : qε (·) −→ q0 (·) as ε → 0, for points s ∈ U[q · (·)], where: (a) q0 (t), t ∈ R+ is a real-valued function from the class L such that (a1) |q0 (t)| ≤ q0,+∫(t), t ≥ 0 for ∞ a non-negative and non-increasing function q0,+ (t), (a2) limT →∞ T q0,+ (s)ds = 0, (b) U[q · (·)] is a Borel subset of R+ = [0, ∞), such that the Lebesgue measure m(U¯ [q · (·)]) = 0, (c) the function q0 (t), t ∈ R+ is continuous almost everywhere with respect to the Lebesgue measure on B+ . It should be noted that in light of the condition Y9 , the majorisation assumption Y10 (a1) for the limiting function q0 (·) looks natural. Lemma A.2 The conditions Y9 and Y10 imply that the conditions Y7 and Y8 are satisfied. Proof The condition Y9 obviously implies that the condition Y7 (a) is satisfied. The following inequality holds for every ε ∈ [0, 1] and T, h > 0, ∫ ∞

h sup |qε (s)| ≤ qε,+ (s)ds. (A.25) r ≥T /h rh ≤s ≤(r+1)h

T −h

The conditions Y9 , Y10 and the relation (A.25) imply that the following relation holds:

376

A Perturbed Renewal Equation

lim

lim h

T →∞ 0≤ε→0

≤ lim



sup

r ≥T /h rh ≤s ≤(r+1)h ∫ ∞

lim

T →∞ 0≤ε→0 T −h

|qε (s)|

qε,+ (s)ds = 0.

(A.26)

Note that the symbol 0 ≤ ε → 0, which admits the case ε ≡ 0, is used in the relation (A.26). Thus, the condition Y7 (b) is satisfied. The condition Y10 (a1) and the relation (A.26), for the case ε ≡ 0, imply that the condition Y8 (a) is satisfied. The asymptotic relations given in the conditions Y10 and Y8 coincide as well as  the conditions Y10 (b), (c), and Y8 (b), (c). The following theorem is a direct corollary of Theorem A.2 and Lemma A.2. Theorem A.3 Let the conditions Y5 , Y6 , Y9 , and Y10 be satisfied. Then, for any 0 ≤ tε → ∞ as ε → 0 such that qε tε → t ∈ [0, ∞] as ε → 0, ∫∞ q0 (s)ds −t/e0, F · 0 as ε → 0. (A.27) xε (tε ) → x0 (∞) = e e0,F

A.2.3 Renewal Theorem for Perturbed Renewal Equation with Transition Renewal Period One of the main areas of application for Theorem A.2 is ergodic theorems for perturbed regenerative- type processes. In the case, where such processes have the so-called transition periods, the following generalisation of Theorem A.2 is useful. Suppose that, for each ε ∈ (0, 1], a function xε (t) from the class L is a solution of the renewal equation (A.18), and xε (t) is a function from the class L connected with the function xε (t) by the following renewal type relation: ∫ t xε (t) = qε (t) + xε (t − s)Fε (ds), t ≥ 0, (A.28) 0

where: (a) qε (t), t ≥ 0 is a real-valued function from the class L, (b) Fε (s) is a proper or improper distribution function on [0, ∞), i.e., Fε (∞) ≤ 1. Let us assume that the following condition is satisfied: Y11 : (a) Fε (·) ⇒ F0 (·) as ε → 0, where F0 (·) is a proper or improper distribution function on [0, ∞), (b) Fε (∞) → F0 (∞) as ε → 0. It is useful to note that the case, where the limiting distribution function F0 (s) = F0 I(s ≥ 0), s ≥ 0, is also allowed. Let also the following condition be satisfied:

A.2 Perturbed Renewal Equation and Renewal Theorem

377

Y12 : lim0≤T →∞ limε→0 supt ≥T | qε (t)| = 0. The following theorem complements Theorem A.2. Theorem A.4 Let the conditions Y5 –Y8 and Y11 –Y12 be satisfied. Then, for any 0 ≤ tε → ∞ as ε → 0 such that qε tε → t ∈ [0, ∞] as ε → 0, ∫∞ q0 (s)ds −t/e0, F · 0 · F0 (∞) as ε → 0. (A.29) xε (tε ) → x0 (∞) = e e0,F Proof The relation (A.28) implies that the following relation holds: ∫ tε xε (tε ) = qε (tε ) + xε (tε − s)Fε (ds) 0 ∫ ∞ = qε (tε ) + xε (tε − s)Fε (ds),

(A.30)

0

where xε (tε − s) = 0, for s > tε . The relation (A.29) given in Theorem A.2 implies, for any tε → ∞ as ε → 0 such that qε tε → t ∈ [0, ∞] as ε → 0 and 0 ≤ sε → s ∈ [0, ∞) as ε → 0, ∫∞ q0 (s)ds as ε → 0. (A.31) xε (tε − sε ) → x0 (∞) = e−t/e0, F · 0 e0,F Indeed, qε · (tε − sε ) → t as ε → 0, and thus, the relation (A.31) is just a variant of the relation (A.24). us The relation (A.31) means that functions xε (tε − ·) −→ x0 (·) as ε → 0, in every point s ∈ [0, ∞), where the function x0 (s) = x0 (∞), s ≥ 0. The relation (A.31) and the condition Y11 imply, by Lemma B.2, that the following relation holds: ∫ ∞ ∫ tε xε (tε − s)Fε (ds) = xε (tε − s)Fε (ds) 0 0 ∫ ∞ → x0 (∞)F0 (ds) = x0 (∞) · F0 (∞) as ε → 0. (A.32) 0

Since tε → ∞ as ε → 0, and, thus, for any T > 0 there exists εT ∈ (0, 1] such that tε ≥ T, for ε ∈ (0, εT ]. This remark and the condition Y12 imply that the following relation holds: lim | qε (tε )| ≤ lim sup | qε (t)| → 0 as T → ∞.

ε→0

ε→0 t ≥T

Finally, the relations (A.32) and (A.33) imply that

(A.33)

378

A Perturbed Renewal Equation

∫ x˜ε (tε ) = qε (tε ) +

0

tε

xε (tε − s)Fε (ds)

→ x0 (∞) · F0 (∞) as ε → 0.

(A.34)

The proof is complete.



The following theorem is a direct corollary of Theorem A.4 and Lemma A.2. Theorem A.5 Let the conditions Y5 , Y6 , and Y9 –Y12 be satisfied. Then, for any 0 ≤ tε → ∞ as ε → 0 such that qε tε → t ∈ [0, ∞] as ε → 0, ∫∞ q0 (s)ds −t/e0, F · 0 · F0 (∞) as ε → 0. (A.35) xε (tε ) → x0 (∞) = e e0,F

Appendix B

Supplementary Asymptotic Results

In this appendix, we present some useful asymptotic results used throughout the book.

B.1 Limit Theorems for Stochastic Processes This section presents some useful limit theorems. In more detail, the relevant material is presented in Appendix A of Volume 1.

B.1.1 Limit Theorems for Randomly Stopped Stochastic Processes Let ξ¯ε (t), t ≥ 0 be, for every ε ∈ [0, 1], a step-sum process with independent increments defined on a probability space Ωε, Fε, Pε ,

(B.1) ξ¯ε,r = (ξε,1 (t), . . . , ξε,k (t)), t ≥ 0, ξ¯ε (t) = r ≤tvε

where: (a) ξ¯ε,r = (ξε,1,r , . . . , ξε,k,r ), r = 1, 2, . . ., are i.i.d. random vectors taking  values in the space Rk , (b) ξε,l (t) = r ≤tvε ξε,l.r , t ≥ 0, for l = 1, . . . , k, and (c) 0 ≤ vε → ∞ as ε → 0. Obviously, ξ¯ε (t), t ≥ 0 is a vector càdlàg stochastic process, i.e., its trajectories are continuous from the right and possess limits from the left at any moment of time. Let also ν¯ε = (νε,1, . . . , νε,k ) be, for every ε ∈ [0, 1], a random vector with non-negative components defined on the same probability space Ωε, Fε, Pε , as the process ξ¯ε (t), t ≥ 0.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0

379

380

B Supplementary asymptotic results

We assume that the following condition holds: d

Z1 : (ν¯ε, ξ¯ε (t)), t ∈ U −→ (ν¯0, ξ¯0 (t)), t ∈ U as ε → 0, where U is some set dense in [0, ∞) and containing point 0. As is known, the limiting process ξ¯0 (t) is, in this case, a Lévy process (càdlàg, homogeneous in time process with independent increments). Necessary and sufficient conditions for the convergence in distribution and in the topology J for processes with independent increments can be found, for example, in Skorokhod (1964, 1986), and Gikhman and Skorokhod (1971). The objects of our interest are random vectors, ζ¯ε = (ζε,1, . . . , ζε,k ) = (ξε,1 (νε,1 ), . . . , ξε,k (νε,k )).

(B.2)

The condition Z1 is not sufficient to provide convergence in distribution for random vectors ζ¯ε . Let us assume that the following continuity condition be satisfied: Z2 : P{ξ0,r (ν0,r ) = ξ0,r (ν0,r − 0)} = 1, for r = 1, . . . , k. The condition Z2 means that the stopping moment ν0,r is a point of continuity for càdlàg process ξ0,r (t) with probability 1, for every r = 1, . . . , k. It is worth noting that the condition Z2 is automatically satisfied, if ξ¯0 (t), t ≥ 0 is a continuous process. Also, the condition Z2 is satisfied, if the process ξ¯0 (t), t ≥ 0 and the random vector ν¯0 are independent. In particular, it is so, in the case where ν¯0 is a.s. a non-random vector. In this case, the condition Z1 can be replaced by two separate assumptions that d d ξ¯ε (t), t ∈ U −→ ξ¯0 (t)), t ∈ U as ε → 0 and ν¯ε −→ ν¯0 as ε → 0. This follows from the Slutsky theorem given below in Sect. B.1.1.2. The following theorem presents a simplified variant of the results obtained in Silvestrov (1971, 1972, 1974, 2004). Theorem B.1 Let the conditions Z1 and Z2 hold. Then, d ζ¯ε −→ ζ¯0 as ε → 0.

(B.3)

B.1.2 Slutsky Theorem and Related Results  , . . . , ξ  ) and ξ¯ = (ξ  , . . . , ξ  ) be, for every ε ∈ [0, 1], random Let ξ¯ε = (ξε,1 ε ε,1 ε,k ε,k vectors defined on a probability space Ωε, Fε, Pε and taking values in the space Rk . Let us assume that the following condition is satisfied: d

d

Z3 : (a) ξ¯ε −→ ξ¯0 as ε → 0. (b) ξ¯ε −→ ξ¯0 as ε → 0, where ξ¯0 is a non-random vector.

B.2 Other Useful Asymptotic Results

381

The following useful theorem takes place. Theorem B.2 Let the condition Z3 be satisfied. Then, d (ξ¯ε , ξ¯ε) −→ (ξ¯0, ξ¯0) as ε → 0.

(B.4)

d The relation (B.4) obviously implies that random variables f (ξ¯ε , ξ¯ε) −→ ¯ y¯ ) acting from Rk × Rk to f (ξ¯0, ξ¯0) as ε → 0, for any continuous function f ( x, ¯ y¯ ) = x¯ + y¯ . Rk . For example, this relation holds for functions f ( x, Let, for ε ∈ [0, 1], the random vector, ξ¯ε+ = ξ¯ε + ξ¯ε. The following theorem follows from Theorem B.2 and the above remark.

Theorem B.3 Let the condition Z3 be satisfied. Then, the following relation holds: d ξ¯ε+ −→ ξ¯0+ as ε → 0. In particular, if the random vector ξ¯0 = 0¯ = (0, . . . , 0), then ξ¯0+ = ξ¯0 . Theorem B.3 has an analogue for stochastic processes.  (t), . . . , ξ  (t)), t ≥ 0 and ξ¯ (t) = (ξ  (t), . . . , ξ  (t)) be, for Let ξ¯ε (t) = (ξε,1 ε ε,1 ε,k ε,k  every ε ∈ [0, 1], vector stochastic processes (with real-valued components) defined on a probability space Ωε, Fε, Pε . The following condition is an analogue of the condition Z3 for stochastic processes. d d Z4 : (a) ξ¯ε (t), t ≥ 0 −→ ξ¯0 (t), t ≥ 0 as ε → 0, (b) ξ¯ε(t), t ≥ 0 −→ ξ¯0(t), t ≥ 0 as ε → 0, where ξ¯0(t), t ≥ 0 is a non-random function.

The following theorem is an analogue of Theorem B.2. Theorem B.4 Let the condition Z4 be satisfied. Then, d

(ξ¯ε (t), ξ¯ε(t)), t ≥ 0 −→ (ξ¯0 (t), ξ¯0(t)), t ≥ 0 as ε → 0.

(B.5)

Suppose that for ε ∈ [0, 1] the stochastic process, ξ¯ε+ (t) = ξ¯ε (t) + ξ¯ε(t), t ≥ 0. The following useful theorem is a direct corollary of Theorem B.4. Theorem B.5 Let the condition Z4 be satisfied. Then, the following relation holds: d ¯ t ≥ 0, then ξ¯ε+ (t), t ≥ 0 −→ ξ¯0+ (t), t ≥ 0 as ε → 0. In particular, if ξ¯0(t) = 0, +  ξ¯0 (t) = ξ¯0 (t), t ≥ 0. The proofs of Theorems B.2–B.5 as well as more general propositions can be found, for example, in Silvestrov (2004).

B.2 Other Useful Asymptotic Results In this section, we present some additional useful asymptotic results used throughout the book.

382

B Supplementary asymptotic results

B.2.1 Asymptotically Comparable Functions Let H = {h(ε)} be a non-empty family of functions h(ε) defined on the interval (0, 1] and taking values in the interval (0, ∞). We call H = {h(·)} a complete family of asymptotically comparable functions if: (1) it is closed with respect to operations of summation, multiplication, and division and (2) there exists limε→0 h(ε) = aH [h(·)] ∈ [0, ∞], for any function h(·) ∈ H. Here, aH [h(·)] is some functional acting from the functional space H to the interval [0, ∞]. It is specific for every complete family of asymptotically comparable functions H. We can refer to aH [h(·)] as a comparability limit for the function h(·). Below we give three typical examples of complete families of asymptotically comparable functions. The first is the family of asymptotically comparable power-type functions H1 = {h(·)} defined on interval (0, 1], taking values in interval (0, ∞) and such that, for any function h(·) ∈ H1 , there exist constants ah > 0 and bh ∈ (−∞, ∞) (comparability parameters for the function h(·)) such that the following asymptotic relation holds: h(ε) → 1 as ε → 0. ah ε bh

(B.6)

The second is the family of asymptotically comparable power-exponential-type functions H2 = {h(·)} defined on interval (0, 1], taking values in interval (0, ∞) and such that, for any function h(·) ∈ H2 , there exist constants ah > 0 and bh, ch ∈ (−∞, ∞) (comparability parameters for the function h(·)) such that the following asymptotic relation holds: h(ε) ah

−1 ε bh e−ch ε

→ 1 as ε → 0.

(B.7)

The third is the family of asymptotically comparable power-logarithmic-type functions H3 = {h(·)} defined on interval (0, 1], taking values in interval (0, ∞) and such that, for any function h(·) ∈ H2 , there exist constants ah > 0 and bh, dh ∈ (−∞, ∞) (comparability parameters for the function h(·)) such that the following asymptotic relation holds: h(ε) → 1 as ε → 0. ah ε bh (1 + ln ε −1 )−dh

(B.8)

The notion of a complete family of asymptotically comparable functions is introduced and commented in detail in Chap. 81 . Also, Chap. 81 contains operating rules that give explicit formulas for calculating the comparability parameters and limits for sums, products, and quotients of asymptotically comparable functions.

B.2 Other Useful Asymptotic Results

383

B.2.2 Convergence of Lebesgue Integrals in the Scheme of Series Let fε (s) be, for every ε ∈ [0, 1], a real-valued Borel function defined on R+ = [0, ∞). Definition B.1 The functions fε (·) converge to the function f0 (·) locally uniformly at the point s ∈ [0, ∞), as ε → 0, if fε (sε ) → f0 (s) as ε → 0, for any 0 ≤ sε → s as ε → 0. us

We use the symbol fε (·) −→ f0 (·) as ε → 0 to denote this convergence. The following lemma clarifies the term of locally uniform convergence. us

Lemma B.1 The functions fε (·) −→ f0 (·) as ε → 0, in some point s ∈ [0, ∞), if and only if the following relation holds, lim lim

sup

0 0, then the condition Z5 (b) implies that με (R+ ) > 0 for ε small enough, say ε ≤ ε0 . The condition Z5 (b) allows us to reduce the proof to the case where με (·) are probability measures. This follows from the identity, which holds for ε ∈ (0, ε0 ], ∫ ∫ fε (s)με (ds) = με (R+ ) fε (s) μ˜ ε (ds), (B.12) R+

μ−1 ε (R+ )με (A)

R+

where μ˜ ε (A) = is a probability measure. Let us choose an arbitrary sequence 0 = ε0 < εn → 0 as n → ∞. It is possible, thanks to the Skorokhod (1956) representation theorem (see, for example, Silvestrov (2004) or Kallenberg (2021)), to construct a sequence of random

384

B Supplementary asymptotic results

variables ηεn defined on a probability space Ω, F, P (the same for all random variables ηεn ) such that: (a) the random variable ηεn has the distribution μεn (·), for a.s. n = 0, 1, . . .; (b) ηεn −→ η0 as n → ∞. Let A be the set of elementary events ω ∈ Ω, such that (c) ηεn (ω) → η0 (ω) as n → ∞. Also, let B be the set of elementary events ω, such that η0 (ω) ∈ S. The statement (b) and the condition Z6 (b) imply that that (d) P(A ∩ B) = 1. The condition Z6 and the relation (c) imply that (e) fεn (ηεn (ω)) → f0 (η0 (ω)) as n → ∞, for ω ∈ A ∩ B. The statements (d) and (e) imply that a.s.

fεn (ηεn ) −→ f0 (η0 ) as n → ∞.

(B.13)

Note that the condition Z6 (a) implies that (f) there exists N < ∞ such that | fεn (ηεn )| ≤ 2 f < ∞ for n ≥ N. Also, the condition Z6 (b) implies that (g) sups ∈S | f0 (s)| ≤ 2 f < ∞, and, thus, (h) | f0 (η0 )| ≤ 2 f < ∞, with probability 1. Hence, by the Lebesgue theorem, it follows from statements (f), (g), and (h) and the relation (B.13) that E fεn (ηεn ) → E f0 (η0 ) as n → ∞. (B.14) Since the limit in (B.14) does not depend on the choice of sequence 0 < εn → 0, the relations (B.12) (in which, as mentioned above, we can assume that με (R+ ) ≡ 1) and (B.14) imply that the relation (B.11) holds.  The proof of Lemmas B.1 and B.2 can be found, for example, in the book by Gyllenberg and Silvestrov (2008). Here, these lemmas are slightly modified for the case, where the corresponding functions and measures are defined on a half-line. The proof of Lemma B.2 is given above, since this lemma is used in the proofs of many theorems presented in the book.

Appendix C

Methodological and Bibliographical Notes

Appendix C contains methodological and bibliographic notes and comments on the new results presented in Volume 2. Some promising issues for future research in this area are also commented on.

C.1 Methodological Notes C.1.1 A Survey of Works Related to the Results Presented in the Book. In this subsection we briefly comment on the works related to the results presented in Volume 2. Volume 2 presents the results of a complete analysis and classification of individual ergodic theorems for perturbed alternating regenerative processes (modulated by regularly or singularly perturbed semi-Markov processes with two states). New types of super-long, long, and short time ergodic theorems are presented for various asymptotic time zones for regularly, singularly, and super-singularly perturbed alternating processes. We also generalise these ergodic theorems on perturbed multialternating regenerative processes modulated by regularly or singularly perturbed finite semi-Markov processes. We use new asymptotic recurrent phase space reduction algorithms for modulating semi-Markov processes, which allow us to embed models of multi-alternating regenerative processes in the model of alternating regenerative processes. Alternating regenerative-type processes and semi-Markov type processes are popular models of stochastic processes that have a variety of applications for queuing, reliability, control, and many other types of stochastic processes and systems. Here, we refer to selected papers and books that contain basic material about these types of stochastic processes and their applications, Smith (1955b), Cox (1962), Feller (1968, 1971), Kingman (1972), Korolyuk, Brodi and Turbin (1974), Çinlar (1974b, 1975), Gikhman and Skorokhod (1973), Cohen (1976), Iglehart and Shedler (1980), Silvestrov (1980a), Volker (1980), Revuz (1984), Shedler (1987, 1993), Kalashnikov (1994b), Korolyuk and Swishchuk (1995a, b), Kovalenko, Kuznetsov Shurenkov © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0

385

386

C Methodological and bibliographical notes

(1996), Kijima (1997), Rolski, Schmidli, Schmidt and Teugels (1999), Thorisson (2000), Limnios and Oprişan (2001), Janssen and Manca (2006, 2007), Barbu and Limnios (2008), Gyllenberg and Silvestrov (2008), Harlamov (2008), Serfozo (2009), Stewart, W. (2009), Collet, Martínez, and San Martín (2013), Ross (2014), Grabski (2015), and Kallenberg (2021). In the case of regularly perturbed models, the corresponding individual ergodic theorems resemble well known ergodic theorems for unperturbed Markov chains, semi-Markov, and regenerative-type processes. First of all, we note the pioneering works of Markov (1906), Kolmogorov (1931, 1937), Doeblin (1936, 1937, 1938, 1940), Lévy (1951), and Harri (1956) on ergodic theorems for Markov chains and works of Cox and Smith (1953). Smith (1954, 1955a, b, 1958), Feller (1961, 1971), and Cox (1962), related to the fundamental renewal theorem, and its applications to ergodic theorem for Markov chains, semi-Markov, and regenerative processes should be mentioned as well as works of Pitman (1974), Griffeath (1975), and Lindvall (1979a), connected with an alternative coupling method for obtaining ergodic theorems for Markov type processes. The works of Smith (1955b), Cox (1962), Feller (1971), Kingman (1972), Cohen (1976), Silvestrov (1980a, 1983, 1984a, b, 1994), Shurenkov (1989, 1998), Kalashnikov (1994b), Kovalenko, Kuznetsov and Shurenkov 1996), Thorisson (2000), Lindvall (2002), Asmussen (2003),Gyllenberg and Silvestrov (2008), Serfozo (2009), Ross (2014), Silvestrov, D. and Silvestrov, S. (2017) represent such ergodic theorems well and aggregate an extensive bibliography on ergodic theorems for unperturbed Markov chains, semi-Markov, and regenerative-type processes. Closer results in the form of uniform (with respect to a certain family of transition characteristics) ergodic theorems for Markov chains, semi-Markov, and regenerative processes can be found in works of Arjas, Nummelin and Tweedie (1978), Kalashnikov (1978, 1994b), Kartashov (1978, 1979, 1982a, b, 1984, 1985a, b, c, 1996b), Shurenkov (1980a, b), Silvestrov (1980b, c, 1983, 1984a, b), Silvestrov and Banakh (1981), Banakh (1982a, b), Thorisson (1983, 1987, 1992, 1998, 2000), Alsmeyer (1986, 1994a, b, 1997), Alimov (1994a, b), Ele˘ıko and Shurenkov (1995b), Roberts, Rosenthal, and Schwartz (1998), Meyn and Tweedie (2009), Blanchet and Zwart (2010). The closest (to ergodic theorems obtained in Volume 2) are ergodic theorems for some classes of perturbed semi-Markov, regenerative, and multi-alternating regenerative processes in the so-called transient (triangular array) mode. In this case, the first thing to mention should be the papers of Silvestrov (1976, 1978, 1979), where the classical renewal theorem, given in Feller (1971), was generalised on the model of the perturbed renewal equation, and Shurenkov (1980a, b), where these results were generalised to a model of the regularly perturbed matrix renewal equation. Some further works in this direction are Silvestrov (1983, 1984a, b, 1994, 1995, 2010, 2014), Kovalenko, Kuznetsov and Shurenkov (1996), Alimov and Shurenkov (1990a, b), Ele˘ıko and Shurenkov (1995a, b), Englund and Silvestrov (1997), Gyllenberg and Silvestrov (2000a, 2008), Englund (2001), Filar, Krieger, and Syed (2002), Yin and Zhang (2005, 2013), Ni, Silvestrov and Malyarenko (2008), Ni (2010a,

C.1 Methodological Notes

387

b, 2011, 2012), Petersson (2013, 2016, 2017), Silvestrov and Petersson (2013), Aliev, Ele˘ıko, and Drebot (2014), and Silvestrov, D., Silvestrov, S., Abola, Biganda, Engström, and Kakuba (2019, 2020b). The principal difference from most of the related results presented in the abovementioned works is that we operate not only with regularly, but also with singularly, perturbed multi-alternating regenerative processes. We propose and use new asymptotic recurrent algorithms for aggregation of regeneration times based on phase space reductions for the corresponding modulating semi-Markov processes. This makes it possible to efficiently compute the corresponding time compression factors that determine the corresponding asymptotic super-long, long, and short time zones and the corresponding limiting stationary distributions in ergodic theorems, as well as to trace effectively the “switching” parameters that determine the shapes of these time compression factors and limiting stationary distributions. The main analytical tool used to obtain ergodic theorems is based on the results concerning generalisation of the renewal theorem to the model of the perturbed renewal equation, given in the works, Silvestrov (1976, 1978, 1979), Gyllenberg and Silvestrov (2008), and quasi-ergodic theorems for perturbed regenerative processes with regenerative lifetimes given in the works, Gyllenberg and Silvestrov (2000a, 2008), Silvestrov (2010, 2014). These methods are combined with limit theorems for the hitting times for regularly and singularly perturbed semi-Markov processes presented in Volume 1. This allows us to obtain the corresponding ergodic theorems under minimal conditions. For example, in the case of unperturbed regenerative processes, these conditions reduce to the minimal conditions of the classical individual ergodic theorem for regenerative processes that directly follows from the famous renewal theorem given in Feller (1971). C.1.2 New Results Presented in the Book. In this section we comment on the new results presented in the book. Part I (Chaps. 2–9) presents ergodic theorems for perturbed alternating regenerative processes modulated by regularly, singularly, or super-singularly perturbed two-state semi-Markov processes. The main achievement is a complete classification of ergodic theorems based on 26 super-long, long, and short time ergodic theorems for perturbed alternating regenerative processes modulated by regularly, singularly, or super-singularly perturbed two-state semi-Markov processes. These theorems are obtained under natural minimal conditions typical for ergodic theorems. They represent an impressive variety of ergodic relations for the distributions of perturbed alternating regenerative processes, for various asymptotic time zones and for various asymptotic communicative structures of phase spaces of modulating semi-Markov processes. Part II (Chaps. 10–13) presents new asymptotic recurrent algorithms for the aggregation regeneration times for multi-alternating regenerative processes modulated by regularly and singularly perturbed finite semi-Markov processes. These algorithms include as an integral part the phase space reduction algorithms for perturbed semi-Markov processes developed and presented in Volume 1. We extend these algorithms to multi-alternating regenerative processes by adding asymptotic recurrent

388

C Methodological and bibliographical notes

algorithms for integral transformation of functions representing free terms in renewal type equations connected with regenerative components. Also, the asymptotic recurrent phase space reduction algorithms for modulating semi-Markov processes based on the total removal of virtual transitions from the trajectories of modulating semi-Markov processes, presented in Volume 1, are supplemented by a more complex modification of such algorithms based on partial removal of virtual transitions from the trajectories of modulating semi-Markov processes. These modified versions of asymptotic recurrent phase space reduction algorithms generate additional admissible variants of asymptotic communicative structures for reduced modulating semi-Markov processes, in particular, for the case of singularly perturbed models. The above algorithms make it possible to construct either embedded alternating regenerative processes with two-state modulating semi-Markov processes or embedded regenerative processes for perturbed multi-alternating regenerative processes with a finite phase space of modulating semi-Markov processes. This allows us to apply ergodic theorems for perturbed alternating regenerative processes and perturbed regenerative processes presented in Part I to the model of perturbed multialternating regenerative processes and obtain new super-long, long, and short time ergodic theorems for regularly and singularly perturbed multi-alternating regenerative processes. Interestingly, in the case of models with a singular perturbation, the embedding procedures based on alternating regenerative processes allow one to obtain a wider range of ergodic theorems than the embedding procedure based on regenerative processes. Part I, which presents a complete classification of ergodic theorems for perturbed alternating regenerative processes modulated by regularly, singularly, and super-singularly perturbed two-state semi-Markov processes, is based on the paper Silvestrov (2018). Various variants of ergodic theorems for regularly perturbed alternating and multialternating regenerative processes can be found in previous works, in particular, in those indicated in Sect. C.1.1. The super-long, long, and short time ergodic theorems for singularly and supersingularly perturbed alternating regenerative processes presented in Part I and singularly and super-singularly perturbed multi-alternating regenerative processes presented in Part II are new results, which have no close analogues in other works in this area of research. C.1.3 New Problems. In this subsection, we formulate some new open problems related to the results presented in Volume 2. First of all, I would like to mention that individual ergodic theorems for onedimensional distributions can be extended to ergodic theorems for time-shifted multivariate distributions of perturbed multi-alternating regenerative processes. In fact, such multivariate distributions are also solutions of systems of renewal type equations, similar to systems of renewal type equations for one-dimensional distributions. It would be interesting to try to give a complete classification of ergodic theorems (based on a more detailed analysis of asymptotic communicative structures for modulating semi-Markov processes) for perturbed multi-alternating regenerative

C.2 General Bibliographical Remarks

389

processes modulated by regularly or singularly perturbed finite semi-Markov processes, similar to the above-mentioned classification of ergodic theorems presented for perturbed alternating regenerative processes in Part I. The next logical step in future research is the generalisation of ergodic theorems for the perturbed multi-alternating regenerative processes presented in the book to the model of perturbed multi-alternating regenerative processes with regenerative lifetimes. There is no doubt that ergodic theorems analogous to those presented in the book can be obtained for perturbed multi-alternating regenerative processes with discrete time. There is also hope that similar ergodic theorems associated with different asymptotic time zones can be obtained for perturbed alternating regenerative processes modulating semi-Markov processes with more general phase spaces. Finally, there is an unbounded field of application of the asymptotic results presented in the book to perturbed queuing, reliability and control models, stochastic networks, and bio-stochastic systems.

C.2 General Bibliographical Remarks In this section, we briefly present a general bibliography of papers in the field of ergodic theorems for perturbed Markov, semi-Markov type, and regenerativetype processes and related problems. We focus on the so-called individual ergodic theorems on the convergence of distributions rather than on ergodic theorems for random averages based on theorems like the law of large numbers and related ergodic theorems for expectations of such averages. The works on individual ergodic theorems can be divided into several groups. C.2.1. In the context of ergodic theorems, we include in this bibliography works on renewal theory and ergodic theorems for regenerative-type processes and Markov type processes as well as rates of convergence in these theorems. A selected sample of references to works on renewal theory and ergodic theorems for regenerative-type processes is Blackwell (1948, 1953), Cox and Smith (1953), Smith (1954, 1955a, b, 1958, 1962), Karlin (1955), Hatori (1959, 1960), Feller (1961, 1968, 1971), Feller and Orey (1961), Cox (1962), Farrell (1962), Garsia and Lamperti (1962/1963), Lamperti (1961, 1962), Heyde (1967), Teugels (1967, 1968), Keilson (1969), Sevast’yanov (1974), Rogozin (1976), Silvestrov (1976, 1978, 1979, 1980a, b, c, 1983, 1984a, b, 2000b, 2014), Lindvall (1977, 1979a, 1982, 1986, 2002), Kalashnikov (1978, 1990, 1994a, b), Kartashov (1978, 1979, 1982a, b, 1984), Silvestrov and Banakh (1981), Banakh, D. (1982a, b), Woodroofe (1982), Thorisson (1983, 1987, 1992, 1995, 1998, 2000), Anichkin (1985), Anderson and Athreya (1987), Gut (1988), Lindvall and Rogers (1996), Englund and Silvestrov (1997), Schmidli (1997), Bertoin (1999), Englund (1999a, b, 2000, 2001), Gyllenberg and Silvestrov (1999b, 2000a, b, 2008), Konstantopoulos and Last (1999), Grey (2001), Fuh (2004b), Glynn and Thorisson (2004), Aliev, Ele˘ıko, and Zabolotskyy (2006),

390

C Methodological and bibliographical notes

Blanchet and Glynn (2007), Ni, Silvestrov and Malyarenko (2008), Blanchet and Zwart (2010), Ni (2010a, b, 2011, 2012), Ekheden and Silvestrov (2011), Afanasyeva and Tkachenko (2013, 2016), Aliev, Ele˘ıko, and Buhrii (2013), Avrachenkov, Piunovskiy, and Zhang (2013, 2018), Afanasyeva and Bashtova (2014), Aliev, Ele˘ıko, and Drebot (2014), Alsmeyer (2014), Borovkov, A. and Borovkov, K. (2014), Ekheden (2014), Athreya, Saha, and Srivastava (2017), Chen, Jian, and Li (2017), Abola, Biganda, Silvestrov, D., Silvestrov, S., Engström, Mango, and Kakuba (2019), Daley and Miyazawa (2019), Silvestrov, D., Silvestrov, S., Abola, Biganda, Engström, Mango, and Kakuba (2019, 2020a, b), Jasiulis-Gołdyn, Misiewicz, Naskre¸t, and Omey (2020), Caputo and Quattropani (2021). C.2.2. Selected works on ergodic theorems and rates of convergence for discrete and continuous time Markov chains are Markov (1906), Kolmogorov (1931, 1937), Doeblin (1936, 1937, 1938, 1940), Lévy (1951), Sevast’yanov (1957), Kingman (1963), Vere-Jones (1962, 1964), Orey (1971), Callaert and Keilson (1973), Pitman (1974), Griffeath (1975, 1976, 1978), Nummelin and Tweedie (1976, 1978), Athreya and Ney (1978a), Keilson (1979, 1998), Nummelin (1978a, 1984), Seneta (1973, 1979, 2006), Loéve (1977), Lindvall (1979b, 2002), Tuominen and Tweedie (1979a, 1979b, 1994), Tweedie (1981, 1983), Nummelin and Tuominen (1982, 1983), Tan Choon Peng (1982, 1983), Janson (1983), Kartashov (1984, 1985a, b, c, 1996a, b, 2000), Revuz (1984), Krengel (1985), van Doorn (1985, 1987, 1991, 2001, 2015), Aldous (1988), Barnsley, Demko, Elton, and Geronimo (1988), Barnsley, Elton, and Hardin (1989), Meyn (1989), Borovkov (1990, 1991, 1998), Semal (1991), Borovkov and Foss (1992, 1994), Meyn and Tweedie (1992, 1993a, b, 1994, 2009), Cox and Greven (1994), Gyllenberg and Silvestrov (1994, 1998, 1999a, 2008), Kalashnikov (1994a, b), Spieksma and Tweedie (1994), Down, Meyn, and Tweedie (1995), Thorisson (1995, 1998, 2000), van Doorn and Schrijner (1995), Lund, Meyn, and Tweedie (1996), Lund and Tweedie (1996), Mengersen and Tweedie (1996), Scott and Tweedie (1996), Stenflo (1996, 1998, 2001), Roberts and Rosenthal (1997), Keilson and Vasicek (1998), Roberts, Rosenthal, and Schwartz (1998), Ambroladze (1999), Roberts and Tweedie (1999), Ambroladze and Wallin (2000), Jarner and Hansen (2000), Billera and Diaconis (2001), Corcoran and Tweedie (2001), Jarner and Tweedie (2001, 2003), Jarner and Roberts (2002, 2007), Borovkov and Hordijk (2004), Fuh (2004a, 2021), Baxendale (2005), Păun (2007), Alsmeyer and Hölker (2009), van Doorn and Pollett (2009), Barbour and Pollett (2010, 2012), Asmussen and Glynn (2011), Glynn (2011a), Oliveira (2012), Kontoyiannis and Meyn (2012), Hervé and Ledoux (2013), Nagaev (2013, 2015), Zhang and Zhu (2013), Borovkov, Decrouez, and Gilson (2014), Glynn and Rhee (2014), Martinez, San Martin, and Villemonais (2014), Shur (2014), Andrieu, Fort, and Vihola (2015), Craiu, Gray, Łatuszyński, Madras, Roberts, and Rosenthal (2015), Jiang (2015), Bertoin and Kortchemski (2016), Betz and Le Roux (2016), Froyland and Stuart (2016), McKinlay and Borovkov (2016), Bansaye and Vatutin (2017), Huang LuJing and Mao Yong-Hua (2017), Jovanovski and Madras (2017), Kaijser (2017), Kamatani (2017), Kulik (2017), Andrieu, Lee, and Vihola (2018), Benaim, Cloez, and Panloup (2018), Rosenthal and Yang (2018), Abola, Biganda, Silvestrov, D., Silvestrov, S., Engström, Mango, and Kakuba (2019), Bierkens, Roberts, and Zitt

C.2 General Bibliographical Remarks

391

(2019), Cordeiro, Kharoufeh, and Oxley (2019), Doukhan and Neumann (2019), Livingstone, Betancourt, Byrne, and Girolami (2019), Silvestrov, D., Silvestrov, S., Abola, Biganda, Engström, Mango, and Kakuba (2019, 2020a, b), Bou-Rabee, Eberle, and Zimmer (2020), Chatterjee and Diaconis (2020), Choi and Huang LuJing (2020), Deng (2020), Devraj, Kontoyiannis, Meyn (2020), Durmus, Guillin, and Monmarché (2020), Grama, Lauvergnat, and Le Page (2020), Arapostathis, Pang, and Zheng (2021), Cardona-Tobón and Palau (2021), Lovas and Rásonyi (2021), Martinez (2021), Wang, Pollock, Roberts, and Steinsaltz (2021). C.2.3. Similar problems for more general semi-Markov processes, alternating regenerative processes with semi-Markov modulation, and other related types of processes were studied in Cheong (1967), Çinlar (1969a, b, 1974a, b, 1975), Cheong and Teugels (1972, 1973), Gikhman and Skorokhod (1973), Kesten (1974), Ezhov and Shurenkov (1976), Korolyuk and Turbin (1976b), Nummelin (1976, 1977, 1978b, 1984), Arjas, Nummelin, and Tweedie (1978), Athreya, McDonald, and Ney (1978), Athreya and Ney (1978b), Ele˘ıko (1980, 1998), Shurenkov (1980a, b, 1983, 1984, 1985, 1989, 1998), Silvestrov (1980a, 1994, 1996a, b, 2000a, 2010, 2018), PezhinskaPosdniakova (1983), Silvestrov and Pezhinska (1984), Alsmeyer (1986, 1994a, b, 1997), Djnaid, Ruzhevich, and Shurenkov (1988), Alimov and Shurenkov (1990a, b), Asmussen (1992), Alimov (1994a, b), Ele˘ıko and Shurenkov (1995a, b), Thorisson (1995, 1998, 2000), Hoppensteadt, Salehi, and Skorokhod (1996), Kartashov (1996b), Stenflo (1996, 1998), Silvestrov and Stenflo (1998), Alsmeyer and Hoefs (2001, 2002), Fuh and Lai (2001), Asmussen and O’Cinneide (2002), Klüppelberg and Pergamenchtchikov (2003), Jara and Komorowski (2011), Fuh (2004b), Fontbona, Guérin, and Malrieu (2012), Avrachenkov, Piunovskiy, and Zhang (2013, 2018), Foss and Zachary (2013), Ivanovs (2014), Aliev, Khaniev, and Gever (2015), Glynn and Haas (2015), Harlamov (2016), Asmussen and Thøgersen (2017), Mitov, and Yanev (2017), Latouche and Simon (2018), Cénac, Chauvin, Noûs, Paccaut, and Pouyanne (2021). C.2.4. Finally, we would also like to mention some books that contain material on individual ergodic theorems for Markov, semi-Markov type, and regenerative-type processes. These are books, Foguel (1969), Feller (1971), Orey (1971), Gikhman and Skorokhod (1973), Silvestrov (1980a), Nummelin (1984), Krengel (1985), Shurenkov (1989, 1998), Kartashov (1996b), Kovalenko, Kuznetsov, and Shurenkov (1996), Thorisson (2000), Häggström (2002), Lindvall (2002), Hernández-Lerma and Lasserre, (2003), Gyllenberg and Silvestrov (2008), Meyn and Tweedie (2009), Collet, Martínez and San Martín (2013), Kulik (2017), Levi and Peres (2017), Silvestrov, D. and Silvestrov, S. (2017), and Kallenberg (2021).

References

1. Abola, B., Biganda, P.S., Silvestrov, D., Silvestrov, S., Engstrröm, C., Mango, J.M., Kakuba, G. (2019). Nonlinearly perturbed Markov chains and information networks. In: Skiadas, C.H. (Ed.) Proceedings of the 18th Applied Stochastic Models and Data Analysis International Conference. Florence, Italy, 2019, ISAST: International Society for the Advancement of Science and Technology, 51–79. 2. Afanasyeva, L.G., Bashtova, E.E. (2014). Coupling method for asymptotic analysis of queues with regenerative input and unreliable server. Queueing Syst., 76, no. 2, 125–147. 3. Afanasyeva, L.G., Tkachenko, A.V. (2013). Multichannel queueing systems with regenerative input flow. Teor. Verojatnost. Primen., 58, no. 2, 210–234 (English translation in Theory Probab. Appl. 58, no. 2, 174–192). 4. Afanasyeva, L.G., Tkachenko, A.V. (2016). On the convergence rate for queueing and reliability models described by regenerative processes. J. Math. Sci., 218, no. 2, 119–136. 5. Aldous, D.J. (1988). Finite-time implications of relaxation times for stochastically monotone processes. Probab. Theory Related Fields, 77, no. 1, 137–145. 6. Aliev, R., Khaniev,T., Gever, B. (2015). Weak convergence theorem for ergodic distribution of stochastic process with a discrete interference of change and generalized reflecting barrier. Teor. Verojatnost. Primen., 60, no. 3, 594–605 (English translation in Theory Probab. Appl., 60, no. 3, 502–513). 7. Aliev, S.A., Ele˘ıko, Ya.I., Buhrii, N.V. (2013). Asymptotical properties of the renewal matrix for some class of infinite-dimensional renewal equations. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 33, no. 4, 3–18. 8. Aliev, S.A., Ele˘ıko, Ya.I., Drebot, A.Y. (2014). Limit theorems and transitory phenomena in the renewal equation in random environments. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 34, no. 1, 11–20. 9. Aliev, S.A., Ele˘ıko, Ya.I., Zabolotskyy, T.N. (2006). Asymptotic behavior of the countable dimensional renewal function. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci., 26, no. 7, 17–26. 10. Alimov, D. (1994a). Limit-ergodic Markov functionals of an ergodic process. Teor. Verojatnost. Primen.. 39, no. 3, 618–626 (English translation in Theory Probab. Appl., 39, no. 3, 504–512). 11. Alimov, D. (1994b). Limit-ergodic Markov functionals of an ergodic process. Teor. Verojatnost. Primen.. 39, no. 4, 657–668 (English translation in Theory Probab. Appl., 39, no. 4, 537–546). 12. Alimov, D., Shurenkov, V.M. (1990a). Markov renewal theorems in triangular array model. Ukr. Mat. Zh., 42, 1443–1448 (English translation in Ukr. Math. J., 42, 1283–1288).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0

393

394

References

13. Alimov, D., Shurenkov, V.M. (1990b). Asymptotic behaviour of terminating Markov processes that are close to ergodic. Ukr. Mat. Zh., 42, 1701–1703 (English translation in Ukr. Math. J., 42 1535–1538). 14. Alsmeyer, G. (1986). Parameter-dependent renewal theorems with applications. Z. Oper. Res., Ser. A-B 30, no. 3, A111–A134. 15. Alsmeyer, G. (1994a). Blackwell’s renewal theorem for certain linear submartingales and coupling. Acta Appl. Math., 34, no. 1-2, 135–150. 16. Alsmeyer, G. (1994b). On the Markov renewal theorem. Stoch. Process. Appl., 50, 37–56 17. Alsmeyer, G. (1997). The Markov renewal theorem and related results. Markov Process. Related Fields, 3, no. 1, 103–127. 18. Alsmeyer, G. (2014). Quasistochastic matrices and Markov renewal theory. J. Appl. Probab., 51A, Celebrating 50 Years of The Applied Probability Trust, 359–376. 19. Alsmeyer, G., Hoefs, V. (2001). Markov renewal theory for stationary m-block factors. Markov Process. Related Fields, 7, no. 2, 325–348. 20. Alsmeyer, G., Hoefs, V. (2002). Markov renewal theory for stationary (m + 1)-block factors: convergence rate results. Stoch. Process. Appl., 98, 77–112. 21. Alsmeyer, G., Hölker, G. (2009). Asymptotic behavior of ultimately contractive iterated random Lipschitz functions. Probab. Math. Statist., 29, no. 2, 321–336. 22. Ambroladze, A. (1999). Ergodic properties of random iterations of analytic functions. Ergodic Theory Dynam. Syst., 19, no. 6, 1379–1388. 23. Ambroladze, A., Wallin, H. (2000). Random iteration of Mobius transformations and Furstenberg’s theorem. Ergodic Theory Dynam. Syst., 20, no. 4, 953–962. 24. Anderson, K.K., Athreya, K.B. (1987). A renewal theorem in the infinite mean case. Ann. Probab., 15, 388–393. 25. Andrieu, C., Fort, G., Vihola, M. (2015). Quantitative convergence rates for subgeometric Markov chains. J. Appl. Probab., 52, no. 2, 391–404. 26. Andrieu, C., Lee, A., Vihola, M. (2018). Uniform ergodicity of the iterated conditional SMC and geometric ergodicity of particle Gibbs samplers. Bernoulli, 24, no. 2, 842–872. 27. Anichkin, S.A. (1985). Rate of convergence estimates in Blackwell’s theorem. In: Stability Problems for Stochastic Models, Varna, 1985. VNIISI, Moscow, 95–102 (English translation in J. Soviet Math., 40, no. 4, 449–453). 28. Arjas, E., Nummelin, E., Tweedie, R.L. (1978). Uniform limit theorems for non-singular renewal and Markov renewal processes. J. Appl. Probab., 15, 112–125. 29. Arapostathis, A., Pang, G., Zheng, Y. (2021). Exponential ergodicity and steady-state approximations for a class of Markov processes under fast regime switching. Adv. Appl. Probab., 53, no. 1, 1–29. 30. Asmussen, S. (1992). On coupling and weak convergence to stationarity. Ann. Appl. Probab., 2, no. 3, 739–751. 31. Asmussen, S. (2003). Applied Probability and Queues. Second edition, Applications of Mathematics, 51, Stochastic Modelling and Applied Probability. Springer, New York, xii+438 pp. 32. Asmussen, S., Glynn, P.W. (2011). A new proof of convergence of MCMC via the ergodic theorem. Statist. Probab. Lett., 81, no. 10, 1482–1485. 33. Asmussen, S., O’Cinneide, C. (2002). On the tail of the waiting time in a Markov-modulated M/G/1 queue. Oper. Res., 50, no. 3, 559–565. 34. Asmussen, S., Thøgersen, J. (2017). Markov dependence in renewal equations and random sums with heavy tails. Stoch. Models, 33, no. 4, 617–632. 35. Athreya, K.B., McDonald, D., Ney, P. (1978). Limit theorems for semi-Markov processes and renewal theory for Markov chains. Ann. Probab. 6, no. 5, 788–797. 36. Athreya, K.B., Ney, P. (1978a). A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc., 245, 493–501. 37. Athreya, K.B., Ney, P. (1978b). Limit theorems for semi-Markov processes. Bull. Austral. Math. Soc., 19, 283–294.

References

395

38. Athreya, K.B., Saha, K., Srivastava, R. (2017). AR(1) sequence with random coefficients: regenerative properties and its application. Commun. Stoch. Anal., 11, no. 3, Article 7, 373–381. 39. Avrachenkov, K., Piunovskiy, A„ Zhang, Y. (2013). Markov processes with restart. J. Appl. Probab., 50, no. 4, 960–968. 40. Avrachenkov, K., Piunovskiy, A„ Zhang, Y. (2018). Hitting times in Markov chains with restart and their application to network centrality. Methodol. Comput. Appl. Probab., 20, no. 4, 1173–1188. 41. Banakh, D. (1982a). Estimates of the rate of convergence in a discrete renewal theorem. Teor. Verojatnost. Mat. Stat., 26, 9–16 (English translation in Theory Probab. Math. Statist., 26, 7–14). 42. Banakh, D. (1982b). Explicit Estimates of the Rate of Convergence for Regenerative Processes with Discrete Time. Candidate of Science dissertation, Kiev State University. 43. Bansaye, V., Vatutin, V. (2017). On the survival probability for a class of subcritical branching processes in random environment. Bernoulli, 23, no. 1, 58–88. 44. Barbour, A.D., Pollett, P.K. (2010). Total variation approximation for quasi-stationary distributions. J. Appl. Probab., 47, no. 4, 934–946. 45. Barbour, A.D., Pollett, P.K. (2012). Total variation approximation for quasi-equilibrium distributions, II. Stoch. Process. Appl. 122, no. 11, 3740–3756. 46. Barbu, V.S., Limnios, N. (2008). Semi-Markov Chains and Hidden Semi-Markov Models Toward Applications. Their Use in Reliability and DNA Analysis. Lecture Notes in Statistics, 191, Springer, New York, xiv+224 pp. 47. Barnsley, M.F., Demko, S.G., Elton, J.H., Geronimo, J.S. (1988). Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities. Ann. Inst. H. Poincaré Probab. Statist., 24, no. 3, 367–394. 48. Barnsley, M.F., Elton, J.H., Hardin, D.P. (1989). Recurrent iterated function systems. Fractal approximation. Constr. Approx., 5, no. 1, 3–31. 49. Baxendale, P.H. (2005). Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Probab., 15, no. 1B, 700–738. 50. Benaim, M., Cloez, B., Panloup, F. (2018). Stochastic approximation of quasi-stationary distributions on compact spaces and applications. Ann. Appl. Probab., 28, no. 4, 2370–2416. 51. Bertoin, J. (1999). Renewal theory for embedded regenerative sets. Ann. Probab., 27, 1523– 1535. 52. Bertoin, J., Kortchemski, I. (2016). Self-similar scaling limits of Markov chains on the positive integers. Ann. Appl. Probab., 26, no. 4, 2556–2595. 53. Betz, V., Le Roux, S. (2016). Multi-scale metastable dynamics and the asymptotic stationary distribution of perturbed Markov chains. Stoch. Process. Appl., 126, no. 11, 3499–3526. 54. Bierkens, J., Roberts, G.O., Zitt, P.A. (2019). Ergodicity of the zigzag process. Ann. Appl. Probab., 29, no. 4, 2266–2301. 55. Blackwell, D. (1948). A renewal theorem. Duke Math. J., 15, 145–150. 56. Blackwell, D. (1953). Extension of a renewal theorem. Pacific J. Math., 3, 315–320. 57. Blanchet, J., Glynn, P. (2007). Uniform renewal theory with applications to expansions of random geometric sums. Adv. Appl. Probab., 39, no. 4, 1070–1097. 58. Blanchet, J., Zwart, B. (2010). Asymptotic expansions of defective renewal equations with applications to perturbed risk models and processor sharing queues. Math. Meth. Oper. Res., 72, 311–326. 59. Borovkov, A.A. (1990). Ergodicity and stability of multidimensional Markov chains. Teor. Verojatnost. Primen., 35, no. 3, 543–547 (English translation in Theory Probab. Appl., 35, no. 3, 542–546). 60. Borovkov, A.A. (1991). Lyapunov functions and the ergodicity of multidimensional Markov chains. Teor. Verojatnost. Primen., 36, no. 1, 93–110 (English translation in Theory Probab. Appl., 36, no. 1, 1–18).

396

References

61. Borovkov, A.A. (1998). Ergodicity and Stability of Stochastic Processes. Wiley Series in Probability and Statistics, 314, Wiley, Chichester, xxiv+585 pp. (Translation from the 1994 Russian original). 62. Borovkov, A.A., Borovkov, K.A. (2014). Analogues of the Blackwell theorem for weighted renewal functions. Sibirsk. Mat. Zh., 55, no. 4, 724–743 (English translation in Sib. Math. J., 55, no. 4, 589–605). 63. Borovkov, A.A., Foss, S.G. (1992). Stochastically recursive sequences and their generalizations. Sib. Adv. Math., 2, no. 1, 16–81. 64. Borovkov, A.A., Foss, S.G. (1994). Two ergodicity criteria for stochastically recursive sequences. Acta Appl. Math., 34, no. 1-2, 125–134. 65. Borovkov, A.A., Hordijk, A. (2004). Characterization and sufficient conditions for normed ergodicity of Markov chains. Adv. Appl. Probab., 36, 227–242. 66. Borovkov, K., Decrouez, G., Gilson, M. (2014). On stationary distributions of stochastic neural networks. J. Appl. Probab., 51, no. 3, 837–857. 67. Bou-Rabee, N., Eberle, A., Zimmer, R. (2020). Coupling and convergence for Hamiltonian Monte Carlo. Ann. Appl. Probab., 30, no. 3, 1209–1250. 68. Callaert, H., Keilson, J. (1973). On exponential ergodicity and spectral structure for birthdeath processes. I, II. Stoch. Process. Appl., 1, 187–216, 217–235. 69. Caputo, P., Quattropani, M. (2021). Mixing time trichotomy in regenerating dynamic digraphs. Stoch. Proces. Appl., 137, 222–251. 70. Cardona-Tobón, N., Palau, S. (2021). Yaglom’s limit for critical Galton-Watson processes in varying environment: A probabilistic approach. Bernoulli, 27, no. 3, 1643–1665. 71. Cénac, P., Chauvin, B., Noûs, C., Paccaut, F., Pouyanne, N. (2021). Variable length memory chains: characterization of stationary probability measures. Bernoulli, 27, no. 3, 2011–2039. 72. Chatterjee, A., Diaconis, P. (2020). Speeding up Markov chains with deterministic jumps. Probab. Theory, Rel. Fields, 178, 1193–1214. 73. Cheong, C.K. (1967). Geometric convergence of semi-Markov transition probabilities. Z. Wahrsch. Verw. Gebiete, 7, 122–130. 74. Cheong, C.K., Teugels, J.L. (1972). General solidarity theorems for semi-Markov processes. J. Appl. Probab., 9, 789–802. 75. Cheong, C.K., Teugels, J.L. (1973). On a semi-Markov generalization of the random walk. Stoch. Process. Appl., 1, 53–66. 76. Chen, J., Jian, S., Li, H. (2017). Representations for the decay parameter of Markov chains. Bernoulli, 23, no. 3, 2058– 2082. 77. Choi, M.C.H., Huang, L.J. (2020). On hitting time, mixing time and geometric interpretations of Metropolis-Hastings reversiblizations. J. Theoret. Probab., 33, no. 2, 1144–1163. 78. Çinlar, E. (1969a). Markov renewal theory. Adv. Appl. Probab., 1, 123–187. 79. Çinlar, E. (1969b). On semi-Markov processes on arbitrary spaces. Proc. Cambridge Philos. Soc., 66, 381–392. 80. Çinlar, E. (1974a). Periodicity in Markov renewal theory. Adv. Appl. Probab., 6, 61–78. 81. Çinlar, E. (1974b). Markov renewal theory: a survey. Manag. Sci., 21, no. 7, 727–752. 82. Çinlar, E. (1975). Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, N.J., x+402 pp. 83. Cohen, J.W. (1976). On Regenerative Processes in Queueing Theory. Lecture Notes in Economics and Mathematical Systems, 121, Springer, Berlin, ix+93 pp. 84. Collet, P., Martínez, S., San Martín, J. (2013). Quasi-Stationary Distributions. Markov Chains, Diffusions and Dynamical Systems. Probability and its Applications, Springer, Heidelberg, xvi+280 pp. 85. Corcoran, J.N., Tweedie, R.L. (2001). Perfect sampling of ergodic Harris chains. Ann. Appl. Probab., 11, 438–451. 86. Cordeiro, J., Kharoufeh, J., Oxley, M. (2019). On the ergodicity of a class of level-dependent quasi-birth-and-death processes. Adv. Appl. Probab., 51, no. 4, 1109–1128. 87. Cox, D.R. (1962). Renewal Theory. Methuen, London and Wiley, New York, ix+142 pp.

References

397

88. Cox, D.R., Smith, W.L. (1953). A direct proof of a fundamental theorem of renewal theory. Skand. Aktuarietidskr., 36, 139–150. 89. Cox, J.T., Greven, A. (1994). Ergodic theorems for infinite systems of locally interacting diffusions. Ann. Probab., 22, no. 2, 833–853. 90. Craiu, R.V., Gray, L., Łatuszyński, K., Madras, N., Roberts, G.O., Rosenthal, J.S. (2015). Stability of adversarial Markov chains, with an application to adaptive MCMC algorithms. Ann. Appl. Probab., 25, no. 6, 3592–3623. 91. Daley, D., Miyazawa, M. (2019). A martingale view of Blackwell’s renewal theorem and its extensions to a general counting process. J. Appl. Probab., 56, no. 2, 602–623. 92. Deng, C.S. (2020). Subgeometric rates of convergence for discrete-time Markov chains under discrete-time subordination. J. Theor. Probab., 33, 522–532. 93. Devraj, A., Kontoyiannis, I., Meyn, S. (2020). Geometric ergodicity in a weighted Sobolev space. Ann. Probab., 48, no. 1, 380–403. 94. Billera, L., Diaconis, P. (2001). A Geometric Interpretation of the Metropolis-Hastings Algorithm. Stat. Sci., 16, no. 4, 335–339. 95. Djnaid, M.O., Ruzhevich, N.A., Shurenkov, V.M. (1988). Potentials of ergodic chains, semiMarkov processes, and sums of nonnegative variables. Teor. Verojatnost. Primen., 33, no. 3, 545–559 (English translation in Theory Probab. Appl., 33, no. 3, 509–522). 96. Doeblin, W. (1936). Sur les chaînes de Markoff. C. R. Acad. Sci. Paris, 203, no. 1, 24–26. 97. Doeblin, W. (1937). Sur deux propiétés asymptotiques de mouvement régis par certains tupes de chaînes simples. Bull. Math. Soc. Raum. Sci., 39, no. 1, 57–115, no. 2, 3–61. 98. Doeblin, W. (1938). Sur deux problems de Kolmogoroff concentrat les chaînes décombrables. Bull. Soc. Math. France, 66, 210–220. 99. Doeblin, W. (1940). Éléments d’une théorie générale des chaînes simples constantes de Markoff. Ann. École Norm., (3) 57, 61–111. 100. Doukhan, P., Neumann, M. (2019). Absolute regularity of semi-contractive GARCH-type processes. J. Appl. Probab., 56, no.1, 91–115. 101. Down, D., Meyn, S.P., Tweedie, R.L. (1995). Exponential and uniform ergodicity of Markov processes. Ann. Probab., 23, no. 4, 1671–1691. 102. Durmus, A., Guillin, A., Monmarché, P. (2020). Geometric ergodicity of the Bouncy Particle Sampler. Ann. Appl. Probab., 30, no. 5, 2069–2098. 103. Ekheden, E., Silvestrov, D. (2011). Coupling and explicit rates of convergence in CramérLundberg approximation for reinsurance risk processes. Commun. Stat. Theory Methods, 40, no. 19-20, 3524–3539. 104. Ekheden, E. (2014). Catastrophe, Ruin and Death - Some Perspectives on Insurance Mathematics. Doctoral dissertation, Stockholm University. 105. Ele˘ıko, Ya.I. (1980). Limit distributions for a semi-Markov process with arbitrary phase space. Teor. Verojatnost. Mat. Stat., 23, 51–58 (English translation in Theory Probab. Math. Statist., 23, 55–61). 106. Ele˘ıko, Ya.I. (1998). Some refinements of limit theorems for semi-Markov processes with a general phase space. Teor. ˇImorvirn. Mat. Stat., 59, 57–65 (English translation in Theory Probab. Math. Statist., 59, 57–65). 107. Ele˘ıko, Ya.I., Shurenkov, V.M. (1995a). Some properties of random evolutions. Ukr. Mat. Zh., 47, 1333–1337 (English translation in Ukr. Math. J., 47, 1519–1525). 108. Ele˘ıko, Ya.I., Shurenkov, V.M. (1995b). Transient phenomena in a class of matrix-valued stochastic evolutions. Teor. ˇImorvirn. Mat. Stat., 52, 72–76 (English translation in Theory Probab. Math. Statist., 52, 75–79). 109. Englund, E. (1999a). Perturbed renewal equations with application to M/M queueing systems. 1. Teor. ˇImovirn. Mat. Stat., 60, 31–37 (Also in Theory Probab. Math. Statist., 60, 35–42). 110. Englund, E. (1999b). Perturbed renewal equations with application to M/M queueing systems. 2. Teor. ˇImovirn. Mat. Stat., 61, 21–32 (Also in Theory Probab. Math. Statist., 61, 21–32). 111. Englund, E. (2000). Nonlinearly perturbed renewal equations with applications to a random walk. In: Silvestrov, D., Yadrenko, M., Olenko A., Zinchenko, N. (Eds.) Proceedings of the Third International School on Applied Statistics, Financial and Actuarial Mathematics, Feodosiya, 2000. Theory Stoch. Process., 6(22), no. 3-4, 33–60.

398

References

112. Englund, E. (2001). Nonlinearly Perturbed Renewal Equations with Applications. Doctoral dissertation, Umeå University. 113. Englund, E., Silvestrov, D.S. (1997). Mixed large deviation and ergodic theorems for regenerative processes with discrete time. In: Jagers, P., Kulldorff, G., Portenko, N., Silvestrov, D. (Eds.) Proceedings of the Second Scandinavian–Ukrainian Conference in Mathematical Statistics, Vol. I, Umeå, 1997. Theory Stoch. Process., 3(19), no. 1-2, 164–176. 114. Ezhov, I.I., Shurenkov, V.M. (1976). Ergodic theorems connected with the Markov property of random processes. Teor. Verojatnost. Primen., 21, 635–639 (English translation in Theory Probab. Appl., 21, 620–623). 115. Farrell, R.H. (1962). Asymptotic renewal theorems in the absolutely continuous case. Duke Math. J., 29, 33–40. 116. Feller, W. (1961). A simple proof for renewal theorems. Comm. Pure Appl. Math., 14, 285– 293. 117. Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Vol. I. Third edition, Wiley, New York, xviii+509 pp. (First edition in 1950). 118. Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II. (Second edition, Wiley Series in Probability and Statistics, Wiley, New York, xxiv+669 pp. (First edition in 1966). 119. Feller, W., Orey, S. (1961). A renewal theorem. J. Math. Mech., 10, 619–624. 120. Filar, J., Krieger, H.A., Syed, Z. (2002). Cesaro limits of analytically perturbed stochastic matrices. Linear Algeb. Appl., 353, no. 1-3, 227–243. 121. Foguel, S.R. (1969). The Ergodic Theory of Markov Processes. Van Nostrand Mathematical Studies, 21, Van Nostrand, New York, 101 pp. 122. Fontbona, J., Guérin, H., Malrieu, F. (2012). Quantitative estimates for the long-time behavior of an ergodic variant of the telegraph process. Adv. Appl. Probab., 44, no. 4, 977–994. 123. Foss, S., Zachary, S. (2013). Stochastic sequences with a regenerative structure that may depend both on the future and on the past. Adv. Appl. Probab., 45, no. 4, 1083-1110. 124. Froyland, G., Stuart, R. (2016). Cheeger inequalities for absorbing Markov chains. Adv. Appl. Probab., 48, no. 3, 631–647. 125. Fuh, C.D. (2004a). Iterated random function system: convergence theorems. In: Probability, finance and insurance, World Sci. Publ., River Edge, NJ, 98–111. 126. Fuh, C.D. (2004b). Uniform Markov renewal theory and ruin probabilities in Markov random walks. Ann. Appl. Probab., 14, no. 3, 1202–1241. 127. Fuh, C.D. (2021). Asymptotic behavior for Markovian iterated function systems, Stoch. Proces. Appl., 138, 186–211. 128. Fuh, C.D., Lai, T.L. (2001). Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks. Adv. Appl. Probab., 33, 652–673. 129. Garsia, A., Lamperti, J. (1962/1963) A discrete renewal theorem with infinite mean. Comment. Math. Helv., 37, 221–234. 130. Gikhman, I.I., Skorokhod, A.V. (1971). Theory of Random Processes. 1. Probability Theory and Mathematical Statistics, Nauka, Moscow, 664 pp. (English edition: The Theory of Stochastic Processes. 1. Fundamental Principles of Mathematical Sciences, 210, Springer, New York (1974) and Berlin (1980)). 131. Gikhman, I.I., Skorokhod, A.V. (1973). Theory of Random Processes. 2. Probability Theory and Mathematical Statistics, Nauka, Moscow, 639 pp. (English edition: The Theory of Stochastic Processes. II. Classics in Mathematics, Springer, Berlin (2004) and Fundamental Principles of Mathematical Sciences, 218, Springer, Berlin (1975)). 132. Glynn, P. (2011a). Wide-sense regeneration for Harris recurrent Markov processes: an open problem. Queueing Syst., 68, no. 3-4, 305–311. 133. Glynn, P.W., Haas, P.J. (2015). On transience and recurrence in irreducible finite-state stochastic systems. ACM Trans. Model. Comput. Simul., 25, no. 4, Art. 25, 19 pp. 134. Glynn, P., Rhee, C. (2014). Exact estimation for Markov chain equilibrium expectations. J. Appl. Probab., 51(A), 377–389.

References

399

135. Glynn, P,W., Thorisson, H. (2004). Limit theory for taboo-regenerative processes. Queueing Syst., 46, no. 3-4, 271–294. 136. Grabski, F. (2015). Semi-Markov Processes: Applications in System Reliability and Maintenance. Elsevier, Amsterdam, xiv+255 pp. 137. Grama, I., Lauvergnat, R., Le Page, É. (2020). Conditioned local limit theorems for random walks defined on finite Markov chains. Probab. Theory, Rel. Fields, 176, 669–735. 138. Grey, D.R. (2001). Renewal theory. In: Stochastic Processes: Theory and Methods. Handbook of Statistics, 19, North-Holland, Amsterdam, 413–441. 139. Griffeath, D. (1975). A maximal coupling for Markov chains. Z. Wahrsch. Verw. Gebiete, 31, 95–106. 140. Griffeath, D. (1976). Coupling Methods for Markov Processes. Ph.D. Thesis, Cornell University, Ithaca. 141. Griffeath, D. (1978). Coupling methods for Markov processes. In: Studies in Probability and Ergodic Theory. Adv. in Math. Suppl. Stud., 2, Academic Press, New York-London, 1–43. 142. Gut, A. (1988). Stopped Random Walks. Limit Theorems and Applications. Applied Probability. A Series of the Applied Probability Trust, 5, Springer, New York, xiv+263 pp. 143. Gyllenberg, M., Silvestrov, D.S. (1994). Quasi-stationary distributions of a stochastic metapopulation model. J. Math. Biol., 33, 35–70. 144. Gyllenberg, M., Silvestrov, D.S. (1998). Quasi-stationary phenomena in semi-Markov models. In: Proceedings of the Second International Symposium on Semi-Markov Models: Theory and Applications, Compiègne, 1998. Univ. Tech. Compiègne, 87–93. 145. Gyllenberg, M., Silvestrov, D.S. (1999a). Quasi-stationary phenomena in semi-Markov processes. In: Janssen, J., Limnios, N. (Eds.) Semi-Markov Models and Applications. Kluwer, Dordrecht, 33–60. 146. Gyllenberg, M., Silvestrov, D.S. (1999b). Cramér-Lundberg and diffusion approximations for nonlinearly perturbed risk processes including numerical computation of ruin probabilities. In: Silvestrov, D., Yadrenko, M., Borisenko, O., Zinchenko, N. (Eds.) Proceedings of the Second International School on Actuarial and Financial Mathematics, Kiev, 1999. Theory Stoch. Process., 5(21), no. 1-2, 6–21. 147. Gyllenberg, M., Silvestrov, D.S. (2000a). Nonlinearly perturbed regenerative processes and pseudo-stationary phenomena for stochastic systems. Stoch. Process. Appl., 86, 1–27. 148. Gyllenberg, M., Silvestrov, D.S. (2000b). Cramér–Lundberg approximation for nonlinearly perturbed risk processes. Insur. Math. Econom., 26, 75–90. 149. Gyllenberg, M., Silvestrov, D.S. (2008). Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems. De Gruyter Expositions in Mathematics, 44, Walter de Gruyter, Berlin, ix+579 pp. 150. Häggström, O. (2002). Finite Markov Chains and Algorithmic Applications. London Mathematical Society Student Texts, 52, Cambridge University Press, 126 pp. 151. Harlamov, B. (2008). Continuous Semi-Markov Processes. Applied Stochastic Methods Series, ISTE, London and Wiley, Hoboken, NJ, 375 pp. 152. Harlamov, B.P. (2016). Final distribution of a diffusion process: semi-Markov approach. Theory Probab. Appl., 60, no. 3, 506–524 (English translation in Theory Probab. Appl., 60, no. 3, 444–459). 153. Harris, T.E. (1956). The existence of stationary measures for certain Markov processes. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. II, Berkeley, CA, 1954–1955. Univ. California Press, Berkeley and Los Angeles, 113– 124. 154. Hatori, H. (1959). Some theorems in an extended renewal theory. I. K¯odai Math. Semin. Repts., 11, no. 3, 139–146. 155. Hatori, H. (1960). Some theorems in an extended renewal theory. II. K¯odai Math. Semin. Repts., 12, no. 1, 21–27. 156. Hernández-Lerma, O. Lasserre, J.B. (2003). Markov Chains and Invariant Probabilities. Progress in Mathematics, 211, Birkhäuser, Basel, xvi+205 pp.

400

References

157. Hervé, L., Ledoux, J. (2013). Geometric ρ-mixing property of the interarrival times of a stationary Markovian arrival process. J. Appl. Probab., 50, no. 2, 598–601. 158. Heyde, C.C. (1967). Asymptotic renewal results for a natural generalization of classical renewal theory. J. Royal Statist. Soc., Ser. B 29, 141–150. 159. Hoppensteadt, F., Salehi, H., Skorokhod, A. (1996). On the asymptotic behavior of Markov chains with small random perturbations of transition probabilities. In: Gupta, A.K. (Ed.) Multidimensional Statistical Analysis and Theory of Random Matrices: Proceedings of the Sixth Eugene Lukacs Symposium. Bowling Green, OH, 1996. VSP, Utrecht, 93–100. 160. Huang L.J., Mao Y.H. (2017). On some mixing times for nonreversible finite Markov chains. J. Appl. Probab., 54, no. 2, 627–637. 161. Iglehart, D.L., Shedler, G.S. (1980). Regenerative Simulation of Response Times in Networks of Queues. Lecture Notes in Control and Information Sciences, 26, Springer, Berlin, xii+204 pp. 162. Ivanovs, J. (2014). Potential measures of one-sided Markov additive processes with reflecting and terminating barriers. J. Appl. Probab., 51, no. 4, 1154–1170. 163. Janson, S. (1983). Renewal theory for M-dependent variables. Ann. Probab. 11, 558–568. 164. Janssen, J., Manca, R. (2006). Applied Semi-Markov Processes. Springer, New York, xii+309 pp. 165. Janssen, J., Manca, R. (2007). Semi-Markov Risk Models for Finance, Insurance and Reliability. Springer, New York, xvii+429 pp. 166. Jara, M., Komorowski, T. (2011). Limit theorems for some continuous-time random walks. Adv. Appl. Probab., 43, no. 3, 782–813. 167. Jarner, S.F., Hansen, E. (2000). Geometric ergodicity of Metropolis algorithms. Stoch, Process. Appl., 85, no. 2, 341–361. 168. Jarner, S.F., Roberts, G.O. (2002). Polynomial convergence rates of Markov chains. Ann. Appl. Probab., 12, no. 1, 224–247. 169. Jarner, S.F., Roberts, G.O. (2007). Convergence of heavy-tailed Monte Carlo Markov chain algorithms. Scand. J. Statist., 34, no. 4, 781–815. 170. Jarner, S.F., Tweedie, R.L. (2001). Locally contracting iterated functions and stability of Markov chains. J. Appl. Probab., 38, 494–507. 171. Jarner, S.F., Tweedie, R.L. (2003). Necessary conditions for geometric and polynomial ergodicity of random-walk-type Markov chains. Bernoulli, 9, no. 4, 559–578. 172. Jasiulis-Gołdyn, B.H., Misiewicz, J.K., Naskre¸t, K., Omey, E. (2020). Renewal theory for extremal Markov sequences of Kendall type, Stoch. Proces. Appl., 130, no. 6, 3277–3294. 173. Jiang, Y. (2015). Mixing time of Metropolis chain based on random transposition walk converging to multivariate Ewens distribution. Ann. Appl. Probab., 25, no. 3, 1581–1615. 174. Jovanovski, O., Madras, N. (2017). Convergence rates for a hierarchical Gibbs sampler. Bernoulli, 23, no. 1, 603–625. 175. Kaijser, T. (2017). A contraction theorem for Markov chains on general state spaces. Rev. Roumaine Math. Pures Appl., 62, no. 2, 355–370. 176. Kalashnikov, V.V. (1978). Qualitative Analysis of the Behaviour of Complex Systems by the Method of Test Functions. Series in Theory and Methods of Systems Analysis, Nauka, Moscow, 247 pp. 177. Kalashnikov, V.V. (1990). Regenerative queueing processes and their qualitative and quantitative analysis. Queueing Systems Theory Appl., 6, no. 2, 113–136. 178. Kalashnikov, V.V. (1994a). Regeneration and general Markov chains. J. Appl. Math. Stoch. Anal., 7, no. 3, 357–371. 179. Kalashnikov, V.V. (1994b). Topics on Regenerative Processes. CRC Press, Boca Raton, FL, 240 pp. 180. Kallenberg, O. (2021). Foundations of Modern Probability. Third Edition. Probability Theory and Stochastic Modelling, 99, Springer, xii+946 pp. 181. Kamatani, K. (2017). Ergodicity of Markov chain Monte Carlo with reversible proposal. J. Appl. Probab., 54, no. 2, 638–654. 182. Karlin, S. (1955). On the renewal equation. Pacific J. Math., 5, 229–257.

References

401

183. Kartashov, N.V. (1978). On explicit estimates of the rate of convergence in the renewal theorem. Teor. Verojatnost. Mat. Stat., 18, 74–79 (English translation in Theory Probab. Math. Statist., 18, 77–82). 184. Kartashov, N.V. (1979). Power estimates for the rate of convergence in the renewal theorem. Teor. Verojatnost. Primen., 24, 600–607 (English translation in Theory Probab. Math. Statist., 24, 606–612). 185. Kartashov, N.V. (1982a). A generalization of Stone’s representation and necessary conditions for uniform convergence in the renewal theorem. Teor. Verojatnost. Mat. Stat., 26, 49–62 (English translation in Theory Probab. Math. Statist., 26, 53–67). 186. Kartashov, N.V. (1982b). Equivalence of uniform renewal theorems and their criteria. Teor. Verojatnost. Mat. Stat., 27, 51–60 (English translation in Theory Probab. Math. Statist., 27, 55–64). 187. Kartashov, N.V. (1984). Criteria for uniform ergodicity and strong stability of Markov chains with a common phase space. Teor. Verojatnost. Mat. Statist. 30, 65–81 (English translation in Theory Probab. Math. Statist., 30, 71–89). 188. Kartashov, N.V. (1985a). Inequalities in stability and ergodicity theorems for Markov chains with a general phase space. I. Teor. Verojatnost. Primen., 30, 230–240 (English translation in Theory Probab. Appl., 30, 247–259). 189. Kartashov, N.V. (1985b). Inequalities in stability and ergodicity theorems for Markov chains with a general phase space. II. Teor. Verojatnost. Primen., 30, 478–485 (English translation in Theory Probab. Appl., 30, 507–515). 190. Kartashov, N.V. (1985c). Asymptotic representations in an ergodic theorem for general Markov chains and their applications. Teor. Verojatnost. Mat. Stat., 32, 113–121 (English translation in Theory Probab. Math. Statist., 32, 131–139). 191. Kartashov, M.V. (1996a). Computation and estimation of the exponential ergodicity exponent for general Markov processes and chains with recurrent kernels. Teor. ˇImovirn. Mat. Stat., 54, 47–57 (English translation in Theory Probab. Math. Statist., 54, 49–60). 192. Kartashov, M.V. (1996b). Strong Stable Markov Chains. VSP, Utrecht and TBiMC, Kiev, 138 pp. 193. Kartashov, M.V. (2000). Calculation of the spectral ergodicity exponent for the birth and death process. Ukr. Mat. Zh., 52, 889–897 (English translation in Ukr. Math. J., 52, 1018–1028). 194. Keilson, J. (1969). On the matrix renewal function for Markov renewal processes. Ann. Math. Statist., 40, 1901–1907. 195. Keilson, J. (1979). Markov Chain Models – Rarity and Exponentiality. Applied Mathematical Sciences, 28, Springer, New York, xiii+184 pp. 196. Keilson, J. (1998). Covariance and relaxation time in finite Markov chains. J. Appl. Math. Stoch. Anal., 11, 391–396. 197. Keilson, J., Vasicek, O.A. (1998). Monotone measures of ergodicity for Markov chains. J. Appl. Math. Stoch. Anal., 11, 283–288. 198. Kesten, H. (1974). Renewal theory for functionals of a Markov chain with general state space. Ann. Probab., 2, 355–386. 199. Kijima, M. (1997). Markov Processes for Stochastic Modelling. Stochastic Modeling Series. Chapman & Hall, London, x+341 pp. 200. Kingman, J.F. (1963). The exponential decay of Markovian transition probabilities. Proc. London Math. Soc., 13, 337–358. 201. Kingman, J.F.C. (1972). Regenerative Phenomena. Wiley Series in Probability and Mathematical Statistics. Wiley, London, xii+190 pp. 202. Klüppelberg, C., Pergamenchtchikov, S. (2003). Renewal theory for functionals of a Markov chain with compact state space. Ann. Probab., 31, no. 4, 2270–2300. 203. Kolmogorov, A.N. (1931). Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann., 104, 415–458 (Russian translation in Uspehi Mat. Nauk, 5, 5–41). 204. Kolmogorov, A.N. (1937). Markov chains with countable number of states. Bul. MGU, Sec. A, Mat. Mekh., 1, no. 3, 1–16.

402

References

205. Konstantopoulos, T., Last, G. (1999). On the use of Lyapunov function methods in renewal theory. Stoch. Process. Appl., 79, 165–178. 206. Kontoyiannis, I., Meyn, S.P. (2012). Geometric ergodicity and the spectral gap of nonreversible Markov chains. Probab. Theory Related Fields, 154, no. 1-2, 327–339. 207. Korolyuk, V.S., Brodi, S.M., Turbin, A.F. (1974). Semi-Markov processes and their application. Probability Theory. Mathematical Statistics. Theoretical Cybernetics, Vol. 11, VINTI, Moscow, 1974, 47–97. 208. Korolyuk, V., Swishchuk, A. (1995a). Semi-Markov Random Evolutions. Mathematics and its Applications, 308, Springer, Dordrecht, x+310 pp. (Revised English edition of Semi-Markov Random Evolutions. Naukova Dumka, Kiev, (1992). 254 pp.). 209. Korolyuk, V.S., Swishchuk, A.V. (1995b). Evolution of Systems in Random Media. CRC Press, Boca Raton, FL, 368 pp. 210. Korolyuk, V.S., Turbin, A.F. (1976b). Asymptotic enlarging of semi-Markov processes with an arbitrary state space. In: Proceedings of the Third Japan-USSR Symposium on Probability Theory, Tashkent, 1975. Lecture Notes in Mathematics, 550, Springer, Berlin, 297–315. 211. Kovalenko, I.N., Kuznetsov, N.Yu., Shurenkov, V.M. (1996). Models of Random Processes. A Handbook for Mathematicians and Engineers. CRC Press, Boca Raton, FL, 446 pp. (A revised edition of the 1983 Russian original). 212. Krengel, U. (1985). Ergodic Theorems. With a supplement by Antoine Brunel. De Gruyter Studies in Mathematics, 6, Walter de Gruyter, Berlin, viii+357 pp. 213. Kulik, A. (2017). Ergodic Behavior of Markov Processes. With Applications to Limit Theorems. De Gruyter Studies in Mathematics, 67, De Gruyter, Berlin, 2017, x+256 pp. 214. Lamperti, J. (1961). A contribution to renewal theory. Proc. Amer. Math. Soc., 12, 724–731. 215. Lamperti, J. (1962). An invariance principle in renewal theory. Ann. Math. Statist., 33, 685– 696. 216. Latouche, G., Simon M. (2018). Markov-modulated Brownian motion with temporary change of regime at level zero. Methodol. Comput, Appl, Probab., 20, no. 4, 1199–1222. 217. Levi, D.A., Peres, Y. (2017). Markov Chains and Mixing Times. Second Edition. Amer. Math. Soc.. Providence, xvi+447 pp. 218. Lévy, P. (1951). Systemes markoviens et stationnares. Gas denombrable. Ann. Sci. Ecole Norm. Sup., 68, 327–381. 219. Limnios, N., Oprişan, G. (2001). Semi-Markov Processes and Reliability. Statistics for Industry and Technology, Birkhäuser, Boston, xii+222 pp. 220. Lindvall, T. (1977). A probabilistic proof of Blackwell’s renewal theorem. Ann. Probab., 5, 482–485. 221. Lindvall, T. (1979a). On coupling of discrete renewal processes. Z. Wahrsch. Verw. Gebiete, 48, 57–70. 222. Lindvall, T. (1979b). A note on coupling of birth and death processes. J. Appl. Probab., 16, 505–512 223. Lindvall, T. (1982). On coupling for continuous time renewal process, J. Appl. Probab., 19, 82–89. 224. Lindvall, T. (1986) On coupling of renewal processes with use of failure rates. Stoch. Process. Appl., 22, 1–15. 225. Lindvall, T. (2002). Lectures on the Coupling Methodol. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, Wiley, New York, xiv+257 pp. (A revised reprint of the 1992 original). 226. Lindvall, T., Rogers, L.C.G. (1996). On coupling of random walks and renewal processes. J. Appl. Probab., 33, 122–126. 227. Livingstone, S., Betancourt, M., Byrne, S., Girolami, M. (2019). On the geometric ergodicity of Hamiltonian Monte Carlo. Bernoulli, 25, no. 4A, 3109–3138. 228. Loève, M. (1977). Probability Theory. I. Fourth edition. Graduate Texts in Mathematics, 45, Springer, New York, xvii+425 pp. (First edition in 1955). 229. Lovas, A., Rásonyi, M. (2021). Markov chains in random environment with applications in queuing theory and machine learning, Stoch. Proces. Appl., 137, 294–326.

References

403

230. Lund, R.B., Meyn, S.P., Tweedie, R.L. (1996). Computable exponential convergence rates for stochastically ordered Markov processes. Ann. Appl. Probab., 6 , no. 1, 218–237. 231. Lund, R.B., Tweedie, R.L. (1996). Geometric convergence rates for stochastically ordered Markov chains. Math. Oper. Res., 21, no. 1, 182–194. 232. Markov, A.A. (1906). Generalisation of the law of large numbers on dependent trials. Izv. Kazan. Fiz.-Mat. Obsch., 15, no. 4, 135–156. 233. Martinez, S. (2021). Entropy of killed-resurrected stationary Markov chains. J. Appl. Probab., 58, no. 1, 177–196. 234. Martinez, S., San Martin, J., Villemonais, D. (2014). Existence and uniqueness of a quasistationary distribution for Markov processes with fast return from infinity. J. Appl. Probab., 51, no. 3, 756–768. 235. McKinlay, S., Borovkov, K. (2016). On explicit form of the stationary distributions for a class of bounded Markov chains. J. Appl. Probab. 53, no. 1, 231–243. 236. Mengersen, K.L., Tweedie, R.L. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist., 24, no. 1, 101–121. 237. Meyn, S.P. (1989). Ergodic theorems for discrete time stochastic systems using a stochastic Lyapunov function. SIAM J. Control Optim., 27, no. 6, 1409–1439. 238. Meyn, S.P., Tweedie, R.L. (1992). Stability of Markovian processes. I. Criteria for discretetime chains. Adv. Appl. Probab., 24, 542–574. 239. Meyn, S.P., Tweedie, R.L. (1993a). Stability of Markovian processes. II. Continuous-time processes and sampled chains. Adv. Appl. Probab., 25, 487–517. 240. Meyn, S.P., Tweedie, R.L. (1993b). Stability of Markovian processes. III. Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Probab., 25, 518–548. 241. Meyn, S.P., Tweedie, R.L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Probab., 4 , no. 4, 981–1011. 242. Meyn, S.P., Tweedie, R.L. (2009). Markov Chains and Stochastic Stability. Cambridge University Press, xxviii+594 pp. (2nd edition of Markov Chains and Stochastic Stability. Communications and Control Engineering Series, Springer, London, 1993, xvi+ 548 pp.). 243. Mitov, K.V., Yanev, N.M. (2017). Limit theorems for some classes of alternating regenerative branching processes. Pliska Stud. Math., 27, 73–90. 244. Nagaev, S.V. (2013). Ergodic theorems for Markov chains with an arbitrary phase space. Dokl. Akad. Nauk, 453, no. 3, 262–264 (English translation in Dokl. Math. 88 (2013), no. 3, 684–686). 245. Nagaev, S.V. (2015). The spectral method and ergodic theorems for general Markov chains. Izv. Ross. Akad. Nauk Ser. Mat., 79, no. 2, 101–136 (English translation in Izv. Math., 79, no. 2, 311–345. 246. Ni, Y. (2010a). Perturbed renewal equations with multivariate nonpolynomial perturbations. In: Frenkel I., Gertsbakh I., Khvatskin L., Laslo Z., Lisnianski A. (Eds.) Proceedings of the International Symposium on Stochastic Models in Reliability Engineering, Life Science and Operations Management. Beer Sheva, Israel, 754–763. 247. Ni, Y. (2010b). Analytical and numerical studies of perturbed renewal equations with multivariate non-polynomial perturbations. J. Appl. Quant. Meth., 5, no. 3, 498–515. 248. Ni, Y. (2011). Nonlinearly Perturbed Renewal Equations: Asymptotic Results and Applications. Doctoral dissertation, 106, Mälardalen University, Västerås. 249. Ni, Y. (2012). Nonlinearly perturbed renewal equations: the non-polynomial case. Teor. ˇImovirn. Mat. Stat., 84, 111–122 (Also in Theory Probab. Math. Statist., 84, 117–129). 250. Ni, Y., Silvestrov, D., Malyarenko, A. (2008). Exponential asymptotics for nonlinearly perturbed renewal equation with non-polynomial perturbations. J. Numer. Appl. Math., 1(96), 173–197. 251. Nummelin, E. (1976). Limit theorems for α-recurrent semi-Markov processes. Adv. Appl. Probab., 8, 531–547. 252. Nummelin, E. (1977). On the concepts of α-recurrence and α-transience for Markov renewal processes. Stoch. Process. Appl., 5, 1–19.

404

References

253. Nummelin, E. (1978a) A splitting techniques for certain Markov chains and its applications. Z. Wahrsch. Verw. Gebiete, 43, 309–318. 254. Nummelin, E. (1978b). Uniform and ratio limit theorems for Markov renewal and semiregenerative processes on a general state space. Ann. Inst. H. Poincaré, Sect. B (N.S.) 14, no. 2, 119–143. 255. Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge Tracts in Mathematics, 83, Cambridge University Press, Cambridge, 170 pp. 256. Nummelin, E., Tuominen, P. (1982), Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory. Stoch. Process. Appl., 12 no. 2, 187–202. 257. Nummelin, E., Tuominen, P. (1983). The rate of convergence in Orey’s theorem for Harris recurrent Markov chains with applications to renewal theory. Stoch. Process. Appl., 15, 295–311. 258. Nummelin, E., Tweedie, R.L. (1976). Geometric ergodicity for a class of Markov chains. In: École d’Été de Calcul des Probabilités de Saint-Flour, Saint-Flour, 1976. Ann. Sci. Univ. Clermont, 61, Math., no. 14 , 145–154. 259. Nummelin, E., Tweedie, R.L. (1978). Geometric ergodicity and R-positivity for general Markov chains. Ann. Probab., 6, 404–420. 260. Oliveira, R.I. (2012). Mixing and hitting times for finite Markov chains. Electron. J. Probab., 17, no. 70, 12 pp. 261. Orey, S. (1971). Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand, London, 108 pp. 262. Păun, U. (2007). Perturbed finite Markov chains. Math. Rep. 9(59), no. 2, 183–210. 263. Petersson, M. (2013). Quasi-stationary distributions for perturbed discrete time regenerative processes. Teor. ˇImovirn. Mat. Stat., 89, 140–155 (Also in Theory Probab. Math. Statist. 89, 153–168). 264. Petersson, M. (2016). Perturbed Discrete Time Stochastic Models. Doctoral dissertation. Stockholm University. 265. Petersson, M. (2017). Quasi-stationary asymptotics for perturbed semi-Markov processes in discrete time. Methodol. Comput. Appl. Probab., 19, 1047–1074. 266. Pezhinska-Posdniakova, G. (1983). Estimates for Rate of Convergence in Ergodic Theorems for Random Processes with Semi-Markov Switchings. Candidate of Science Dissertation, Kiev State University. 267. Pitman, J.W. (1974). Uniform rates of convergence for Markov chain transition probabilities. Z. Wahrsch. Verw. Gebiete, 29, 193–227. 268. Revuz, D. (1984). Markov Chains. North-Holland Mathematical Library, 11, North-Holland, Amsterdam, Elsevier, New York, 374 pp. (Revised version of 1st 1975 edition). 269. Roberts, G.O., Rosenthal, J.S. (1997). Shift-coupling and convergence rates of ergodic averages. Comm. Statist. Stoch. Models, 13, no. 1, 147–165. 270. Roberts, G.O., Rosenthal, J.S., Schwartz, P.O. (1998). Convergence properties of perturbed Markov chains. J. Appl. Probab., 35, 1–11. 271. Roberts, G.O., Tweedie, R.L. (1999). Bounds on regeneration times and convergence rates for Markov chains. Stochastic Process. Appl., 80, no. 2, 211–229. 272. Rogozin, B.A. (1976). Asymptotic analysis of the renewal function. Teor. Verojatnost. Primen., 21, 689–706 (English translation in Theory Probab. Appl., 21, 669–686). 273. Rolski, T., Schmidli, H., Schmidt, V., Teugels, J. (1999). Stochastic Processes for Insurance and Finance. Wiley Series in Probability and Statistics, Wiley, New York, xviii+654 pp. 274. Rosenthal, J.S., Yang, J. (2018). Ergodicity of combocontinuous adaptive MCMC algorithms. Methodol. Comput. Appl. Probab. 20, 535–551. 275. Ross, S.M. (2014). Introduction to Probability Models. Eleventh edition. Elsevier/Academic Press, Amsterdam, xvi+767 pp. 276. Schmidli, H. (1997). An extension to the renewal theorem and an application to risk theory. Ann. Appl. Probab., 7, 121–133.

References

405

277. Scott, D.J., Tweedie, R.L. (1996). Explicit rates of convergence of stochastically ordered Markov chains. In: Athens Conference on Applied Probability and Time Series Analysis, Vol. I, Athens, 1995. Lecture Notes in Statistics, 114, Springer, New York, 176–191. 278. Semal, P. (1991). Iterative algorithms for large stochastic matrices. Linear Algebra Appl., 154/156, 65–103 279. Seneta, E. (1973). Nonnegative Matrices. An Introduction to Theory and Applications. Wiley, New York, x+214 pp. 280. Seneta, E. (1979). Coefficients of ergodicity: structure and applications. Adv. Appl. Probab., 11, 576–590. 281. Seneta, E. (2006). Nonnegative Matrices and Markov chains. Springer Series in Statistics, Springer, New York, 2006, xvi+287 pp. (A revised reprint of 2nd edition of Nonnegative Matrices and Markov Chains. Springer Series in Statistics. Springer, New York, 1981, xiii+279 pp.). 282. Serfozo, R. (2009). Basics of Applied Stochastic Processes. Probability and Its Applications. Springer, Berlin, xiv+443 pp. 283. Sevast’yanov, B.A. (1957). An ergodic theorem for Markov processes and its application to telephone systems with refusals. Teor. Verojatnost. Primen., 2, no.1, 106–116 (English translation in Theory Probab. Appl., 2, no.1, 104–112). 284. Sevast’yanov, B.A. (1974). Renewal Theory. In: Itogi Mauki i Tehniki. Ser. Teor. Verojatnost. Mat. Stat. Teor. Kibernet., 11, 99–128 (English translation in J. Sov. Math., 1975, 4:3, 281– 302). 285. Shedler, G.S. (1987). Regeneration and Networks of Queues. Applied Probability. A Series of the Applied Probability Trust. Springer, New York, viii+224 pp. 286. Shedler, G.S. (1993). Regenerative Stochastic Simulation. Statistical Modeling and Decision Science. Academic Press, Boston, MA, x+400 pp. 287. Shur, M.G. (2014). Two theorems on convergence parameter of an irreducible Markov chain. Teor. Verojatnost. Primen., 58, no.1, 200–205 ( English translation in Theory Probab. Appl., 58, no. 1, 159–164). 288. Shurenkov, V.M. (1980a). Transition phenomena of the renewal theory in asymptotical problems of theory of random processes 1. Mat. Sbornik, 112, 115–132 (English translation in Math. USSR: Sbornik, 40, no. 1, 107–123). 289. Shurenkov, V.M. (1980b). Transition phenomena of the renewal theory in asymptotical problems of theory of random processes 2. Mat. Sbornik, 112, 226–241 (English translation in Math. USSR: Sbornik, 40, no. 2, 211–225). 290. Shurenkov, V.M. (1983). Final probabilities of ergodic Markov processes. In: Itô, K., Prokhorov, Yu.V. (Eds). Probability Theory and Mathematical Statistics. Proceedings of the Fourth USSR–Japan Symposium, Tbilisi, 1982. Lecture Notes in Mathematics, 1021, Springer, Berlin, 655–665. 291. Shurenkov, V M. (1984). On Markov renewal theory. Teor. Verojatnost. Primen., 29, no. 2, 248–263 (English translation in Theory Probab. Appl., 29, no. 2, 247– 265). 292. Shurenkov, V.M. (1985). Markovian interference of chance and limit theorems. Mat. Sb. 126(168) , no. 2, 172–193 (English translation in Math. USSR-Sb., 54, no. 1, 161–183). 293. Shurenkov, V.M. (1989). Ergodic Markov Processes. Probability Theory and Mathematical Statistics, Nauka, Moscow, 332 pp. 294. Shurenkov, V.M. (1998). Ergodic Theorems and Related Problems. VSP, Utrecht, viii+96 pp. (English edition of: Ergodic Theorems and Related Problems of Theory of Random Processes. Naukova Dumka, Kiev, 119 pp. (1981)). 295. Silvestrov, D.S. (1971). Limit distributions for compositions of random functions. Dokl. Akad. Nauk SSSR, 199, 1251–1252 (English translation in Soviet Math. Dokl., 12, 1282–1285). 296. Silvestrov, D.S. (1972). Remarks on the limit of composite random function. Teor. Verojatnost. Primen., 17, no. 4, 707–715 (English translation in Theory Probab. Appl., 17, no. 4, 669–677). 297. Silvestrov, D.S. (1974). Limit Theorems for Composite Random Functions. Vysshaya Shkola and Izdatel’stvo Kievskogo Universiteta, Kiev, 318 pp.

406

References

298. Silvestrov, D.S. (1976). A generalization of the renewal theorem. Dokl. Akad. Nauk Ukr. SSR, Ser. A, no. 11, 978–982. 299. Silvestrov, D.S. (1978). The renewal theorem in a series scheme.1. Teor. Verojatnost. Mat. Stat., 18, 144–161 (English translation in Theory Probab. Math. Statist., 18, 155–172). 300. Silvestrov, D.S. (1979). The renewal theorem in a series scheme.2. Teor. Verojatnost. Mat. Stat., 20, 97–116 (English translation in Theory Probab. Math. Statist., 20, 113–130). 301. Silvestrov, D.S. (1980a). Semi-Markov Processes with a Discrete State Space. Library for the Engineer in Reliability, Sovetskoe Radio, Moscow, 272 pp. 302. Silvestrov, D.S. (1980b). Synchronized regenerative processes and explicit estimates for the rate of convergence in ergodic theorems. Dokl. Acad. Nauk Ukr. SSR, Ser. A, no. 11, 22–25. 303. Silvestrov, D.S. (1980c). Explicit estimates in ergodic theorems for regenerative processes. Elektron. Inform. Kybern., 16, no. 8-9, 461–463. 304. Silvestrov, D.S. (1983). Method of a single probability space in ergodic theorems for regenerative processes 1. Math. Operat. Statist., Ser. Optim. 14, 285–299. 305. Silvestrov, D.S. (1984a). Method of a single probability space in ergodic theorems for regenerative processes 2. Math. Operat. Statist., Ser. Optim. 15, 601–612. 306. Silvestrov, D.S. (1984b). Method of a single probability space in ergodic theorems for regenerative processes 3. Math. Operat. Statist., Ser. Optim. 15, 613–622. 307. Silvestrov, D. (1994). Coupling for Markov renewal processes and the rate of convergence in ergodic theorems for processes with semi-Markov switchings. Acta Applic. Math. 34, 109–124. 308. Silvestrov, D.S. (1995). Exponential asymptotic for a perturbed renewal equation. Teor. ˘Imovirn. Mat. Stat., 52, 143–153 (English translation in Theory Probab. Math. Statist., 52, 153–162). 309. Silvestrov, D.S. (1996a). Hitting times for semi-Markov dynamical systems. Theory Stoch. Process., 2(18), no. 1-2, 260–268. 310. Silvestrov, D.S. (1996b). Recurrence relations for generalised hitting times for semi-Markov processes. Ann. Appl. Probab., 6, 617–649. 311. Silvestrov, D.S. (2000a). Nonlinearly perturbed Markov chains and large deviations for lifetime functionals. In: Limnios, N., Nikulin, M. (Eds.) Recent Advances in Reliability Theory: Methodology, Practice and Inference. Birkhäuser, Boston, 135–144. 312. Silvestrov, D.S. (2000b). The perturbed renewal equation and diffusion type approximation for risk processes. Teor. ˘Imovirn. Mat. Stat., 62, 134–144 (English translation in Theory Probab. Math. Statist., 62, 145–156). 313. Silvestrov D.S. (2004). Limit Theorems for Randomly Stopped Stochastic Processes. Probability and Its Applications, Springer, London, xvi+398 pp. 314. Silvestrov D.S. (2010). Nonlinearly perturbed stochastic processes and systems. In: Rykov, V., Balakrishnan, N., Nikulin, M. (Eds). Mathematical and Statistical Models and Methods in Reliability. Birkhäuser, Chapter 2, 19–38. 315. Silvestrov D.S. (2014). Improved asymptotics for ruin probabilities. In: Silvestrov, D., MartinLöf, A. (Eds). Modern Problems in Insurance Mathematics, Chapter 5, EAA series, Springer, Cham, 93–110. 316. Silvestrov, D. (2018). Individual ergodic theorems for perturbed alternating regenerative processes. In: Silvestrov, S., Ran˘cić, M., Malyarenko, A. (Eds.) Stochastic Processes and Applications, Chapter 3, Springer Proceedings in Mathematical Statistics, 271, Springer, Cham, 23–89. 317. Silvestrov, D.S., Banakh, D. (1981). Uniform mean ergodic theorems for accumulation processes. Dokl. Acad. Nauk Ukr. SSR, Ser. A, no. 2, 36–38. 318. Silvestrov, D.S., Petersson, M. (2013). Exponential expansions for perturbed discrete time renewal equations. In: Karagrigoriou, A., Lisnianski, A., Kleyner, A., Frenkel, I. (Eds.) Applied Reliability Engineering and Risk Analysis. Probabilistic Models and Statistical Inference. Chapter 23, Wiley, 349–362.

References

407

319. Silvestrov, D.S., Pezhinska, G. (1984). Explicit estimates for the rate of convergence in ergodic theorems for random processes with semi-Markov switchings. In: Random Analysis and Asymptotical Problems of Probability Theory and Mathematical Statistics. Mitsniereba, Tbilisi, 72–78. 320. Silvestrov, D., Silvestrov, S. (2017). Nonlinearly Perturbed Semi-Markov Processes. Springer Briefs in Probability and Mathematical Statistics, Springer, Cham, xiv+143 pp. 321. Silvestrov, D., Silvestrov, S., Abola,B., Biganda, P.S., Engström, C., Mango, J.M., Kakuba, G. (2019). Coupling and ergodic theorems for Markov chains with damping component. Teor. Verojatnost. Mat. Stat., 101, 212–231 (Also in Theory Probab. Math. Statist, 101, 243–264). 322. Silvestrov, D., Silvestrov, S., Abola, B., Biganda, P.S., Engstrröm, C., Mango, J.M., Kakuba, G. (2020a). Perturbation analysis for stationary distributions of Markov chains with damping component. In: Silvestrov, S., Ran˘cić, M., Malyarenko, A. (Eds.) Algebraic Structures and Applications, Chapter 38. Springer Proceedings in Mathematics & Statistics, 317, Springer, Cham, 903–934. 323. Silvestrov, D., Silvestrov, S., Abola, B., Biganda, P.S., Engstrröm, C., Mango, J.M., Kakuba, G. (2020b). Perturbed Markov chains with damping component. Methodol. Comput, Appl, Probab., doi.org/10.1007/s11009-020-09815-9, 31 pp., 23, no. 1 (2021), 369–397. 324. Silvestrov, D.S., Stenflo, Ö. (1998). Ergodic theorems for iterated function systems controlled by regenerative sequences. J. Theor. Probab., 11, 589–608. 325. Skorokhod, A.V. (1956). Limit theorems for stochastic processes. Teor. Verojatnost. Primen., 1, 289–319 (English translation in Theory Probab. Appl., 1, 261–290). 326. Skorokhod, A.V. (1964). Random Processes with Independent Increments. Probability Theory and Mathematical Statistics, Nauka, Moscow, 278 pp. (English edition: Nat. Lending Library for Sci. and Tech., Boston Spa, 1971). 327. Skorokhod, A.V. (1986). Random Processes with Independent Increments. Second edition, Probability Theory and Mathematical Statistics, Nauka, Moscow, 320 pp. (English edition: Mathematics and its Applications, 47, Kluwer, Dordrecht, 1991, xii+279 pp.). 328. Smith, W.L. (1954). Asymptotic renewal theorems. Proc. Roy. Soc. Edinburgh., Sect. A. 64, 9–48. 329. Smith, W.L. (1955a). Extensions of a renewal theorem. Proc. Cambridge Philos. Soc., 51, 629–638. 330. Smith, W.L. (1955b). Regenerative Stochastic Processes. Proceedings of the Royal Society, Ser. A: Mathematical, Physical and Engineering Sciences, 232, 6–31. 331. Smith, W.L. (1958). Renewal theory and its ramifications. J. Royal Statist. Soc., Ser. B 20, 243–302. 332. Smith, W.L. (1962). On necessary and sufficient conditions for the convergence of the renewal density. Trans. Amer. Math. Soc., 104, no. 1, 79-100. 333. Spieksma, F.M., Tweedie, R.L. (1994). Strengthening ergodicity to geometric ergodicity for Markov chains. Comm. Statist. Stoch. Models, 10, no. 1, 45–74. 334. Stenflo, Ö. (1996). Iterated function systems controlled by a semi-Markov chain. In: Klesov, O., Korolyuk, V., Kulldorff, G., Silvestrov, D. (Eds) Proceedings of the First Ukrainian– Scandinavian Conference on Stochastic Dynamical Systems, Uzhgorod, 1995. Theory Stoch. Process., 2(18), no. 1-2, 305–315. 335. Stenflo, Ö. (1998). Ergodic Theorems for Iterated Function Systems Controlled by Stochastic Sequences. Ph.D. Thesis, Umeå University. 336. Stenflo, Ö. (2001). Ergodic theorems for Markov chains represented by iterated function systems. Bull. Polish Acad. Sci. Math., 49, no. 1, 27–43. 337. Stewart, W.J. (2009). Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling. Princeton University Press, Princeton, NJ, xviii+758 pp. 338. Tan, C.P. (1982). A functional form for a particular coefficient of ergodicity. J. Appl. Probab., 19, 858–863. 339. Tan, C.P. (1983). Coefficients of ergodicity with respect to vector norms. J. Appl. Probab., 20, no. 2, 277–287.

408

References

340. Teugels, J.L. (1967). Exponential decay in renewal theorems. Bull. Soc. Math. Belg., 19, 259–276. 341. Teugels, J.L. (1968). Renewal theorems when the first or the second moment is infinite. Ann. Math. Statist., 39 1210–1219. 342. Thorisson, H. (1983). The coupling of regenerative processes. Adv. Appl. Probab., 15, 531– 561. 343. Thorisson, H. (1987). A complete coupling proof of Blackwell’s renewal theorem. Stoch. Process. Appl., 26, 87–97. 344. Thorisson, H. (1992). The coupling method and regenerative processes. In: Analysis, Algebra, and Computers in Mathematical Research, Luleå, 1992. Lecture Notes in Pure and Applied Mathematics, 156, Dekker, New York, 347–363. 345. Thorisson, H. (1995). Coupling methods in probability theory. Scand. J. Statist., 22, no. 2, 159–182. 346. Thorisson, H. (1998). Coupling. In: Probability Towards 2000, New York, 1995. Lecture Notes in Statistics, 128, Springer, New York, 319–339. 347. Thorisson, H. (2000). Coupling, Stationarity and Regeneration. Probability and its Applications, Springer, New York, xiv+517 pp. 348. Tuominen, P., Tweedie, R.L. (1979a). Exponential ergodicity in Markovian queueing and dam models. J. Appl. Probab., 16, no. 4, 867–880. 349. Tuominen, P., Tweedie, R.L. (1979b). Exponential decay and ergodicity of general Markov processes and their discrete skeletons. Adv. Appl. Probab., 11, no. 4, 784–803. 350. Tuominen, P., Tweedie, R.L. (1994). Subgeometric rates of convergence of f-ergodic Markov chains. Adv. in Appl. Probab., 26, no. 3, 775–798. 351. Tweedie, R.L. (1981). Criteria for ergodicity, exponential ergodicity and strong ergodicity of Markov processes. J. Appl. Probab., 18, 122–130. 352. Tweedie, R.L. (1983) Criteria for rates of convergence of Markov chains, with application to queueing and storage theory. In: Probability, Statistics and Analysis. London Math. Soc. Lecture Note Ser., 79, Cambridge Univ. Press, Cambridge-New York, 260–276. 353. van Doorn, E.A. (1985). Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process. Adv. Appl. Probab., 17, 514–530. 354. van Doorn, E.A. (1987). The indeterminate rate problem for birth-death processes. Pacific J. Math., 130, no. 2, 379–393. 355. van Doorn, E. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Probab., 23, 683–700. 356. van Doorn, E.A. (2001). Representations for the rate of convergence of birth-death processes. Teor. ˇImovirn. Mat. Stat., 65, 33–38 (Also in Theory Probab. Math. Statist., 65, 37–43). 357. van Doorn, E. (2015). Representations for the decay parameter of a birth-death process based on the Courant-Fischer theorem. J. Appl. Probab., 52, no. 1, 278–289. 358. van Doorn, E.A., Pollett, P.K. (2009). Quasi-stationary distributions for reducible absorbing Markov chains in discrete time. Markov Process. Related Fields, 15, no. 2, 191–204. 359. van Doorn, E.A., Schrijner, P. (1995). Geometric ergodicity and quasi-stationarity in discretetime birth-death processes. J. Austral. Math. Soc., Ser. B 37, no. 2, 121–144. 360. Vere-Jones, D. (1962). Geometric ergodicity in denumerable Markov chains. Quart. J. Math., 13, 7–28. 361. Vere-Jones, D. (1964). A rate of convergence problem in the theory of queues. Teor. Verojatnost. Primen., 9, 104–112 (Also in Theory Probab. Appl., 9, 94–103). 362. Volker, N. (1980). Semi-Markov Processes. Scientific Paperbacks, Mathematics/Physics Series, 260. Akademie-Verlag, Berlin, 162 pp. 363. Wang, A.Q., Pollock, M., Roberts, G.O., Steinsaltz, D. (2021). Regeneration-enriched Markov processes with application to Monte Carlo. Ann. Appl. Probab., 31, no. 2, 703–735. 364. Woodroofe, M. (1982). Nonlinear Renewal Theory in Sequential Analysis. CBMS-NSF Regional Conference Series in Applied Mathematics, 39, Soc. Industr. Appl. Math., Philadelphia, v+113 pp.

References

409

365. Yin, G.G., Zhang, Q. (2005). Discrete-Time Markov Chains. Two-Time-Scale Methods and Applications. Stochastic Modelling and Applied Probability, Springer, New York, xix+348 pp. 366. Yin, G.G., Zhang, Q. (2013). Continuous-Time Markov Chains and Applications. A TwoTime-Scale Approach. Stochastic Modelling and Applied Probability, 37. Springer, New York, xxii+427 pp. (2nd revised edition of Continuous-Time Markov Chains and Applications. A Singular Perturbation Approach. Applications of Mathematics, 37, Springer, New York, 1998. xvi+349 pp.). 367. Zhang, H., Zhu, Y. (2013). Domain of attraction of the quasistationary distribution for birthand-death processes. J. Appl. Probab., 50, no. 1, 114–126.

Index

C Class Γ, 35, 36, 51 Convergence almost sure (a.s), 384 asymptotic uniform, 34 locally uniform, 383 weak, 373, 383 Convolution, 368 n-fold, 367 D Decomposition Lebesgue, 370 Distribution one-dimensional, 42 stationary, 38 E Equation perturbed renewal, 372 renewal, 28, 37, 367 F Factor first time compression, 136 second time compression, 136 time compression, 58, 67, 135, 180, 361 Family of asymptotically comparable functions, 12 of asymptotically comparable powerexponential-type functions, 382 of asymptotically comparable powerlogarithmic-type functions, 382

of asymptotically comparable power-type functions, 382 complete of asymptotically comparable functions, 12, 382 Function absolutely continuous distribution, 370 arithmetic distribution, 369 directly Riemann integrable, 371 discrete distribution, 370 non-arithmetic distribution, 30, 31, 369 normalisation, 15 renewal, 368 singular distribution, 370 strictly arithmetic distribution, 369 tail probability, 32, 372 weakly non-arithmetic distribution, 369 G Graph transition, 11, 13 I Index random, 115 switching, 44, 188 L Lifetime modified regenerative, 40 regenerative, 6, 28 shifted regenerative, 29 Limit comparability, 382

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 D. Silvestrov, Perturbed Semi-Markov Type Processes II, https://doi.org/10.1007/978-3-030-92399-0

411

412 M Markov chain ergodic, 55 Measure renewal, 368 Mode failure, 2, 12 partial work, 12 work, 2, 12 Model regularly perturbed, 8 semi-regularly perturbed, 8 singularly perturbed, 8, 58 super-singularly perturbed, 58 Moment of change of state, 66 of state change, 109 P Parameter time scaling, 180 Period transition, 29, 65, 174, 191 Perturbation regular, 192 singular, 192 Probability limiting, 181 stationary, 75, 79, 80, 86, 123, 167, 339, 344, 351, 353, 357, 360 stopping, 30 Procedure of partial removal of virtual transitions, 256 of total removal of virtual transitions, 256 Process absolutely singular perturbed alternating regenerative, 59 alternating regenerative, 7 càdlàg, 379 compressed regenerative, 129 embedded regenerative of the first type, 70 embedded regenerative of the second type, 108 embedded regenerative process, 65 ergodic semi-Markov, 75, 79 limiting alternating regenerative, 106 limiting regenerative, 36 Markov renewal, 2 Markov renewal with partly removed virtual transitions, 221 modulated alternating regenerative, 44, 62, 64, 188 modulating semi-Markov, 7 perturbed regenerative, 42

Index regenerative, 6, 28 regenerative with transition period, 29, 75 regularly perturbed alternating regenerative, 55, 69, 70, 79 regularly perturbed standard alternating regenerative, 74 semi-Markov, 2, 44 semi-Markov with removed virtual transitions, 221 semi-regularly perturbed alternating regenerative, 85 shifted regenerative, 29 singularly perturbed alternating regenerative, 56, 107, 108 standard regenerative, 28 step-sum with independent increments, 379 stochastic measurable, 27 super-singularly perturbed alternating regenerative, 56, 57, 163, 164 unperturbed regenerative, 42 Property continuity, 182 loss-memory, 6 R Relation majorization, 34 renewal type transition, 30 S Solution of renewal equation, 368 Space phase, 27 probability, 27, 43, 188 Span, 369 Stability asymptotic, 57 Sum random, 115 System forward Kolmogorov, 5, 17, 128, 349 queuing, 2, 11 regularly perturbed queuing, 4 of renewal type equations, 45, 52, 196, 287, 294, 302, 309 singularly perturbed queuing, 4 T Theorem individual ergodic, 42, 55, 56, 58 long time ergodic, 4, 58, 129, 167 quasi-ergodic, 42 renewal, 372, 374

Index short time ergodic, 4, 58, 135, 137, 139, 142, 146, 170, 354 Skorokhod representation, 383 Slutsky, 381 super-long time ergodic, 4, 58, 123, 165 Time aggregated regeneration, 64, 65 change of state, 65 degenerating regeneration, 93 first hitting, 65 first return, 80 regeneration, 28, 44, 62, 64, 65, 188 return, 65 sojourn, 44

413 stopping, 65, 201 Transform Laplace, 47, 81 Transformation, 17 time compression, 61 steady, 5 Triplet stochastic, 28, 44 Z Zone asymptotic time, 58, 59, 136, 180 equivalent asymptotic time, 59, 136