Probability Theory and Mathematical Statistics: Proceedings of the Seventh Vilnius Conference, Vilnius, Lithuania, 12–18 August, 1998 [Reprint 2020 ed.] 9783112314081, 9783112302811

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Probability Theory and Mathematical Statistics

Probability Theory and Mathematical Statistics Proceedings of the Seventh Vilnius Conference (1998)

Vilnius, Lithuania, 12-18 August, 1998

Edited by B. Grigelionis, J. Kubilius, V. Paulauskas H. Pragarauskas, R. Rudzkis and V. Statulevicius

Vilnius, Lithuania

III VSP III

Utrecht, The Netherlands

VSP BV P.O. Box 346 3700 AH Zeist The Netherlands

TEV Ltd. Akademijos 4 2600 Vilnius Lithuania

©VSP BV & TEV Ltd. 1999

First published in 1999

ISBN 90-6764-313-0 (VSP) ISBN 9986-546-72-9 (TEV)

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Typeset in Lithuania by TEV Ltd., Vilnius, SL 1185 Printed in Lithuania by Spindulys, Kaunas

CONTENTS

Foreword

ix

Rate of convergence in the transference max-limit theorem A. Aksomaitis

1

Theorems of large deviations for non-Gaussian approximation A. Aleskeviciene and V. Statulevicius

5

Algorithm for calculation of joint distribution of bootstrap sample elements A. Andronov and M. Fioshin

15

Bismut type differentiation of semigroups M. Arnaudon and A. Thalmaier

23

Random permutations and the Ewens sampling formula in genetics G. J. Babu and E. Manstavicius

33

On large deviations of self-normalized sum A. Basalykas

43

Minimax estimation in the blurred signal model E. Belitser

57

Invariant measures of diffusion processes in Hilbert spaces and Hilbert manifolds Y. Belopolskaya

67

One term Edgeworth expansion for Student's t statistic M. Bloznelis and H. Putter

81

Asynchronous stochastic approximation and adaptation in a competitive system R. T. Buche and H. J. Kushner

99

Homogenization of random parabolic operator with large potential F. Campillo, M. Kleptsyna and A. Piatnitski

115

Remarks on infinitely divisible approximations to the binomial law V. Cekanavicius

135

Rates of convergence to stable and discrete stable laws G. Christoph and K. Schreiber

147

Normal-ordered white noise differential equations. II: Regularity properties of solutions Dong Myung Chung, Un Cig Ji and Nobuaki Obata

157

VI

Contents

Line models for snowflake crystals and their Hausdorff dimension J. Croft

175

Multiplicative strong unimodality revisited I. Cuculescu and R. Theodorescu

187

Random walks with jumps in random environments (examples of cycle and weight representations) Y. Derriennic

199

Analysis of generating functions and probabilities on trees M. Drmota

213

On the distribution of roots of polynomials in sectors. Ill A. Dubickas

229

Asymptotic expansions for the moments of the regret risk of classification based on mixed variables K. Ducinskas

235

On the structure of strong Markov continuous local Dirichlet processes H. J. Engelbert and J. Wolf

241

On minimax robustness of Bayesian statistical prediction V. Galinskij and A. Kharin

259

On the zeros of the Lerch zeta-function. II R. Garunkstis

267

The Galton-Watson tree conditioned on its height J. Geiger and G. Kersting

277

Prognosis of experiment results, based on a two-stage examination and linking the inverse Polya distribution to Bayes' rule T. Gerstenkorn

287

Applications of semiparametric biased sampling models to AIDS vaccine and treatment trials P. B. Gilbert and R. J. Bosch

295

Bayesian estimation of common long-range dependent models Nan-Jung Hsu, B. K. Ray and F. J. Breidt

311

Starting an image segmentation: a level set approach J. Istas and M. Hoebeke

325

Wide-ranging interpretations of the cycle representations of Markov processes S. Kalpazidou

339

Analysis of price, income and money demand indicators B. Kaminskiene

355

Robustness in cluster analysis of multivariate observations Yu. Kharin and E. Zhuk

363

Contents

vii

An elementary approach to filtering in systems with fractional Brownian observation noise M. L. Kleptsyna, A. Le Breton andM.-C.

Roubaud

373

Bootstrap tests for parametric volatility structure in nonparametric autoregression J. -P. Kreiss and M. H. Neumann

393

Value-distribution of general Dirichlet series A. Laurincikas

405

The Robbins-Monro type SDE and recursive estimation N. Lazrieva andT. Toronjadze

415

A squared binomial tree approach to discrete-time bond market modelling R. Leipus

429

Strong consistency of k-centres in reflexive spaces J. Lember and K. Pàrna

441

Asymptotic behavior of small ball probabilities M. A. Lifshits

453

Good statistical practice in analytical chemistry P. Lischer

469

On the principle of maximum entropy on the mean as a methodology for the regularization of inverse problems P. Maréchal

481

The rate of convergence in the central limit theorem for endomorphisms of two-dimensional torus G. Misevicius

493

Simple methods for the analysis of multivariate and longitudinal categorical data G. Molenberghs and L. Danielson

499

Trimmed sums and associated random variables in the q-domain of attraction of stable laws D. Neuenschwander

and R. Schott

515

p-Variation and integration of sample functions of stochastic processes R. Norvaisa

521

Functional central limit theorems on Lie groups. A survey G. Pap

541

On some new results for cointegrated processes with infinite variance innovations V. Paulauskas

553

viii

Contents

Sampling methods in Lithuanian official statistics. Design and parameter estimation problems A. Plikusas

571

On ¿''-estimates of stochastic integrals H. Pragarauskas

579

Volatility estimation and bootstrap Z Praskova

589

Random fields and central limit theorem in some generalized Holder spaces A. Rackauskas and Ch. Suquet

599

Consistent estimation of discriminant space in mixture model by using projection pursuit M. Radavicius and R. Rudzkis

617

Nonparametric wavelet methods for nonstationary time series R. von Sachs

627

Some results on arithmetical functions W. Schwarz

653

Linear approximation of random processes and sampling design problems 0. V. Seleznjev

665

Convergence of solution of stochastic equations in the space of formal series I. Ya. Spectorsky

685

Unified inference in survey sampling 1. Traat and K. Meister

697

Semiclassical analysis and a new result in excursion theory A. Truman and I. M. Davies

701

On KPP equations with various wave front speeds, the Larson result via large deviations A. Yu. Veretennikov

707

Co-ordinating and monitoring the transformation process in statistics in emerging democracies and economies in transition - the Polish experience T. Walczak

715

Branching diffusions with random offspring distributions A. Wulfsohn

727

FOREWORD

The 7 t h Vilnius Conference on Probability Theory and Mathematical Statistics was held 12-18 August 1999 in conjunction with the 22 n d European Meeting of Statisticians. The joint conference was organized by European Regional Committee of the Bernoulli Society, Institute of Mathematics and Informatics, Vilnius Gediminas Technical University, Vilnius University, the Lithuanian Academy of Sciences, the Lithuanian Mathematical Society, and the Lithuanian Statistical Society. The Organizing Committee was headed by Professor Vytautas Statulevicius and Chairman of the Programme Committee was Professor Peter Jagers. It was the first time that the Vilnius Conference and the European Meeting of Statisticians were held together. This provided a good opportunity for meetings and discussions between representatives from theoretical branches of probability and statistics, and those working in applied probability and practical statistics. Furthermore, the joint conference reflected new trends in applications, such as financial mathematics, official statistics, and many more. Participants from 37 countries attended the conference - numerous from countries traditionally present, such as Russia, USA, Germany, Great Britain and France, but also participants from non-traditional countries such as Cyprus, Mongolia and Kuwait were attending. The conference was divided into 34 sessions, during which 350 reports were delivered, including approximately 100 invited lectures and 5 plenary lectures. The conference was opened by the President of the Republic of Lithuania, Valdas Adamkus. The plenary lectures were given by Professors R. Gill (opening lecture), N. Reid (forum lecture), J.F. Le Gall (closing lecture), C.D. Cutler (IMS lecture), V. Bentkus (special Vilnius lecture). As a rule, 3 invited speakers were chosen by the session organizers for each session. The Proceedings contain the papers by invited speakers submitted to the Organizing Committee, as well as some selected papers by contributing speakers. We would like to offer our sincere thanks to all the participants who have contributed to the success of the conference. On behalf of the organizers we would also like to express our special gratitude to the scientific secretary of the conference, Dr. Aleksandras Plikusas. We hope that these Proceedings will promote a further advance in probability theory and mathematical statistics. The Editors

Prob. Theory and Math. Stat., pp. 1 - 4 B. Grigelionis et al. (Eds) © 1999 VSP/TEV

RATE OF CONVERGENCE IN THE TRANSFERENCE MAX-LIMIT THEOREM A. AKSOMAITIS Kaunas University of Technology, Studentu 50-325, 3028 Kaunas, Lithuania

ABSTRACT Here we present a nonuniform estimate of the convergence rate in the transference theorem in maxscheme for independent and identically distributed random variables. Presented results give more exact the convergence rate that obtained by the author earlier.

1. INTRODUCTION Let {Xj, j > 1} be a sequence of independent and identically distributed random variables (r.v.'s) with a distribution function F(x) = P(Xj ^ x), j ^ 1, and {G„,n ^ 1} be a sequence of continuous strongly increasing functions defined in the interval (I, s) of values taken by the r.v.'s X j , j ^ 1. After normalizing the r.v.'s Xj we get Xn} = G~\Xj), j > 1. Let [Nn,n > 1} be a sequence of positive integral-valued r.v.'s independent of the X j , j > 1. We form quantities n Mn = \ / X j=1

n j

,

Nn MNn = \ J x n j . 7=1

In the case when Gn(x) = xbn + an (an e R, bn > 0), the necessary and sufficient conditions of weak convergence P(Ai„ i j x) => H(x), n oo, to nonsingular distribution function H are presented in the fundamental work of Gnedenko (1943). Nonuniform estimate of convergence rate \V(Mn ^x)-H(x)\

^ An(x)

(1.1)

was obtained in monograph (Galambos, 1978). More general estimates of convergence rate are given in the paper (Aksomaitis, 1988). We are interested in the nonuniform estimates of convergence rate in the transference theorem (Gnedenko and Gnedenko, 1984), i.e. estimates of the type ÀNn(x)

= |P(M,v„ si x) -

(1.2)

2

A. Aksomaitis

here, distribution function ^(JC) = Hz(x) dA(z) and A(x) = lim„^OO A„(nx) = lim„_>.oo P { N n / n ^ *). In partial case (linear normalizing), using estimate (1.1), the estimate A N „ ( X ) is known (Aksomaitis, 1987). This estimate was improved in the paper (Aksomaitis, 1988). Our aim can be formulated as follows: to get estimates of nonuniform convergence rate in the transference theorem provided estimates \P(Mn^x)-H(x)\^A^(x), \Hn{nx)

(1.3)

- A{x)\ < À