Probability Theory and Mathematical Statistics: Proceedings of the Sixth Vilnius Conference, Vilnius, Lithuania, 28 June–3 July, 1993 [Reprint 2020 ed.] 9783112319321, 9783112308158

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Probability Theory and Mathematical Statistics

Probability Theory and Mathematical Statistics Proceedings of the Sixth Vilnius Conference

(1993)

Vilnius, Lithuania, 28 June - 3 July, 1993

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

T®¥

Vilnius, Lithuania

IIIVSPIII

Utrecht, The Netherlands

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

TEV Ltd. Akademijos 4 Vilnius Lithuania

© VSP BV/TEV Ltd. 1994 First published in 1994 ISBN 90-6764-178-2

AH 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.

CIP-DATA KONINKLIJKE BIBLIOTHEEK, DEN HAAG Probability Probability theory and mathematical statistics: proceedings of the sixth Vilnius Conference / ed. by B. Grigelionis ... [et al.]. Vilnius : Mokslas ; Utrecht: VSP Conference organized by the Institute of Mathematics and Informatics and Vilnius University from June 28-July 3, 1993. ISBN 90-6764-178-2 (VSP) bound NUGI811 Subject headings: probability theory / mathematical statistics.

Typeset in Lithuania by TEV Ltd., Vilnius Printed in The Netherlands by Koninklijke Wöhrmann bv, Zutphen.

CONTENTS

Preface

ix

Large deviations in approximation by Poisson law A. AleSkeviiiene and V. StatuleviSius Limits of weak interaction for stochastic particle system R.R. Ahmitzyanov

1 19

Rate of convergence in the limit transfer theorem of the maximum and minimum A. Aksomaitis

35

Geometry of the state spaces in quantum probability ShA. Ayupov and NJ. Yadgorov

43

From diffusions processes with jumps L. Beznea and L.toStoica

53

Central limit theorem in Skorohod spaces and asymptotic strength distribution of fiber bundles M. Bloznelis and V. Paulauskas

75

Ergodicity in a sense of weak convergence, equilibrium-type identities and large deviations for Markov chains A.A. Borovkov and D. Korshunov

89

On second-order properties of mixing random sequences and random fields R.C. Bradley and SA. Utev

99

Optimal couplings and application to Riemannian geometry Mu-Fa Chen

121

Empirical and partial sum processes with sample path in Banach function spaces M. Csdrgd and R. NorvaiSa

143

Applications of multi-time parameter processes to change-point analysis M. Csdrgd and B. Szyszkowicz

159

vi

Contents

Bergstròm-type asymptotic expansions in the first uniform Kolmogorov's theorem v: ¿ekanavidius

223

Differentiation of heat semigroups and applications K.D. Elworthy and X. -M. Li

239

On the realisation of classical Markov processes as quantum flows in Fock space F. Fagnola

253

Unimodular eigenvalues and invariant probabilities for some classes of linear operators E. Flytzanis and L. Kanakis

277

On function-indexed partial-sum processes P. Gaenssler and K. Ziegler

285

Canonical spectral equation for the eigenvalues of empirical covariance matrices V.L. Girko

313

Conditionally exponential families and Lundberg exponents of Markov additive processes B. Grigelionis

337

On the nonuniform approximation by means of distributions of sums of conditionally independent random variables P. Gudynas

351

Kolmogorov's strong law of large numbers: the amart point of view B. Heinkel

361

Cycle processes S. Kalpazidou

369

Density-dependent branching processes as randomly perturbed iterations of maps F.C. Klebaner

391

Stability of the inverse Radon transform L.B. Klebanov and AA. Zinger

401

On conditions for existence of solutions of integral equations with stochastic line integrals A.M. Kolodii

405

Contents

vii

Nonlinear transformations of empirical processes: functional inverses and Bahadur-Kiefer representations V. Koltchinskii

423

Rate of convergence in the functional central limit theorem for one-dimensional semimartingales K. Kubilius

447

On limit theorems for the Riemann zeta-function in some spaces A. Lauriniikas

457

A posteriori and sequential methods of change-point detection in FARIMA-type time series R. Leipus

485

Limit theorems for the local density of measures in the hypotheses testing problems of counting processes Yu.N. Lin'kov

497

On the strong independence of sequential ranks S.V. Malov

517

A proof of the Erdfis arcsine law E. Manstaviiius

533

Phase transitions divisors M. Mendds Franceandand G. Tenenbaum On the existence of a viscosity solution to the integro-differential Bellman equation

541

R. MikuleviZius and H. Pragarauskas

553

Quasi-state decompositions for quantum spin systems B. Nachtergaele

565

OnNagaev accuracy of approximation with stable laws S.

591

Distributions of processes governed by hyperbolic equations when also boundary conditions are assumed E. Orsingher

605

Central limit theorems on stratified Lie groups G. Pap

613

On the convergence rate in the multi-dimensional martingale CLT A. Raikauskas

629

viii

Contents

Stein-Chen method for compound Poisson approximation: the coupling approach M. Roos

645

Some aspects of the development of probabilistic number theory W. Schwarz

661

Mixture of extremal distributions - a model alternative to the log-linear model E.-M. Tiit

703

Infinite-dimensional M-estimation A.W. van der Vaart

715

On large deviations in averaging principle for systems of stochastic differential equation with unbounded coefficients A.Yu. Veretennikov

735

PREFACE The traditional Sixth International Vilnius Conference on Probability Theory and Mathematical Statistics was held on June 28 - July 3,1993. It was organized by the Institute of Mathematics and Informatics and Vilnius University. The Organizing Committee was headed by Professor V. Statulevicius; the International Program Committee by Professor J. Kubilius. In comparison to previous conferences, this conference was considerably reduced in size due to the decision to restrict the number of parallel sessions. Two hundred and forty scientists from 32 countries participated in the conference. The main subjects of the conference were: limit theorems, stochastic analysis and stochastic physics, quantum probability theory, statistics, change detection in random processes, probabilistic number theory. Professor J.A. Wellner (USA) and Professor L. Accardi (Italy) delivered plenary lectures. One hundred and ninety reports including 50 invited lectures were made at 32 sectional sessions. The Poster session included 27 communications. The present Proceedings of the Conference contain papers by invited speakers submitted to the Organizing Committee as well as some selected papers by other participants. We would like to offer our sincere thanks to all participants who contributed to the success of the Sixth Vilnius Conference. We hope the Proceedings will promote a further advance in the probability theory and mathematical statistics. B. Grigelionis, J. Kubilius, H. Pragarauskas, V. Statulevicius

Prob. Theory and Math. Stat., pp. 1 - 1 8 R. Grigclionis et at. (Eds)

© 1994 v s p m i v

LARGE DEVIATIONS IN APPROXIMATION BY POISSON LAW A . A L E ä K E V I Ö I E N E and V. S T A T U L E V l C l U S Institute of M a t h e m a t i c s and Informatics, A k a d e m i j o s 4, 2 6 0 0 Vilnius, Lithuania Vytautas M a g n u s University, K a u n a s , Lithuania

ABSTRACT Let X be a random variable (r.v.), assuming non-negative integer values, EX = A > 0 and the factorial cumulants of which satisfy some conditions of growth. Then under the condition (S) (see below) a general lemma (Lemma 1) is proved comparing the behaviour of probabilities of large deviations P{X x} of a r.v. X against Poisson distribution P{t] ^ x}, where jj is a Poisson r.v. with parameter A > 0 and A < x < (l/6e)AA. Under the condition (S') the probability P{X > x} is estimated for all x > A (Lemma 2). Several papers have appeared recently (cf. Wolf and Mikosch, 1985; AleSkevicien6, 1988; D e h e u v e l s , 1992; C h e n L o u i s a n d Choi, 1992), in w h i c h large deviation t h e o r e m s for s u m s S„ = + . . . + X ^ of independent in each series random variables x\n), i = 1 , 2 , . . . , are studied, and in w h i c h the usual normal approximation for s u m Sn is replaced by a Poisson approximation. In (Statulevicius and AleSkevicienfe, 1993) w e h a v e also investigated the probabilities of large deviations in a p p r o x i m a t i o n by the Poisson law. B u t instead of sum Sn w e have studied a r a n d o m variable X , the factorial c u m u l a n t s of w h i c h satisfy s o m e g r o w t h conditions. O f course, instead of X w e can t a k e a sum S^ of a series of independent or d e p e n d e n t random variables. T h e present w o r k is an extension of (Statulevicius and AleSkevicienfe, 1993) w h e r e the general l e m m a of large deviations w a s proved (i.e. for a random variable X ) in fact only in the interval A < i < £(AA) 2 / 3 ,

0 < i < 1,

A=

EX.

N o w w e prove this l e m m a in the interval A < x < SAA. (This interval is optional in the sense of order A). In addition, w e have obtained a m o r e exact estimate of the r e m a i n d e r term in this l e m m a . A s m e n t i o n e d , by virtue of the proved l e m m a and the results obtained for estimation of factorial c u m u l a n t s (see (Statulevicius and AleSkevicienfe, 1993; Sidoravicius and Statulevicius, 1991)), o n e can obtain

2

A. AleSkeviiiene

and

V.

Statuleviiius

theorems of large deviations for sums of series of independent or dependent random variables, and some statistics. One can find examples how to do that in (Statuleviiius and AleSkevicienfe, 1993). Furthermore, it ought to be noted that the proof in (Statulevicius and AleSkevicienfe, 1993) was based on the method of factorial cumulants which is, in a sense, an analog of ordinary cumulants, used by one of the authors and his colleagues (Statulevicius, 1966; Rudzkis et al., 1978; Saulis and Statulevicius, 1991) and other papers in to prove theorems of large deviations with a usual normal approximation. In this work we also will use the method of factorial cumulants, further developing it. Let a random variable (r.v.) X assume non-negative integer values. If EXk < oo then factorial moments and cumulants of the Jfc-th order of the r.v. X are defined as follows:

EX(k) = EX (X - 1)... (X - k + 1), =

£ v=\

ki+...+

*•

EX(ki)...EX(K).

k„ = k

In special case, when rj is a Poisson r.v. with parameter A, Er}(k) = Er1(r,-\)...(r}-k+\) and F

= Xk,

^ ) = {o;

k=

1,2,... ,

£ > a

Denote zi = zi(ii) = e " — 1. If for some integer 5 > 0 the factorial moment EX( 3 ) exists (i.e. EX(4) < oo), then = e ( i + * l ( i t))x = £ *=o and

+o(i*r)

(i)

_ logiSe"* = ¿ i ^ z f O O + o(|