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Volume I

Probability Theory and Mathematical Statistics Proceedings of the Fifth Vilnius Conference June 25 - July 1,1989

Edited by

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B. Grigelionis, Yu. V. Prohorov, V. V. Sazonov and V. Statulevicius

MOKSLAS Vilnius, Lithuania

IIIVSPIII

Utrecht, The Netherlands

VSP BV Post Box 346 3700 AH Zeist The Netherlands

MOKSLAS Zvaigzdziu 23 Vilnius Lithuania

©1990 VSP BV/IMI Lithuanian Ac. Sei.

First published in 1990

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 owners.

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CIP-DATA KONINKLIJKE BIBLIOTHEEK, D E N HAAG Probability theory and mathematical statistics: Proceedings of the Fifth Vilnius Conference: Vilnius, Lithuania, J u n e 25 - J u l y 1, 1989 ed. by B . Grigelionis... [et al.] - Utrecht: V S P B V / Vilnius: Mokslas. I S B N 90-6764-128-6 (vol.1) I S B N 90-6764-130-8 (set) S I S O 517 U D C 519.2(063) Subject headings: probability theory/ mathematical statistics.

Typeset in Lithuania by Baltic Amadeus / Publishing Service Group of IMI, Vilnius Printed in Lithuania by Spindulys, Kaunas

CONTENTS Preface

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The low density limit and the quantum Poisson process L. Accardi and Lu Yun Gang

vii 1

Systems assisting in gathering and analysis of statistical data and adjustable to subject domains S.A. Aivazyan, Yu.N. Blagoveschensky and L.D. Meshalkin

28

On the maximum of sums of random variables and the supremum for stable processes A.K. Aleskeviciene

35

On the necessity of Cramer's condition in local limit theorems N.N. Amosova

52

Markov random graphs and polygonal fields with Y-shaped nodes T. AraJc and D. Surgailis

57

Probability measures on quantum logics and a non-commutative Choquet theory Sh.A. Ayupov

68

Convolution semigroups of instruments in quantum probability A. Barchielli and G. Lupieri

78

Stochastic posterior equations for quantum nonlinear filtering V.P. Belavkin

91

Diffusions and parabolic equations in principal bundles Ya.I. Belopolskaya

110

On Markov processes associated with a projective sequence of harmonic spaces A.D. Bendikov

118

Estimation of the tail of the spectral distribution by means of high level sojourn times S.M. Berman

128

Renormalization of Dyson's vector-valued hierarchical model at low temperatures P.M. Bleher and P. Major

141

Analitic functionals of stochastic processes and infinite dimensional oscillatory integrals V.I. Bogachev

152

iv

Contents

Exponential inequalities for the distributions of von Mises and (/-statistics I.S. Borisov

166

Ergodicity and stability of Markov chains and of their generalizations. Multidimensional chains A.A. Borovkov

179

Event and time averages: stationary case P. Bremaud

189

The stationary case and the non-

On the convergence of sums of independent random vectors normed by matrices V.V. Buldygin and S.A. Solntsev

197

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C L T for families of integral functionals arising in solving the multidimensional Burgers Equation

A.V. Bulinskii

207

The theory of dynamical semigroups and its applications A.M. Chebotarev

217

An approximation of integer-valued measures by generalized Poisson measures V. Cekanavicius

228

Lim inf results for the standardized empirical distribution function E. Csaki

238

The smooth measures in Banach spaces and their smooth mappings Yu.L. Daletsky and V.R. Steblovskaya

245

Local invariance principle for i.i.d. random variables Yu.A. Davydov

258

Conditionally positive definite functions on the coefficient algebra of a nonabelian group L.V. Denisov and A.S. Holevo

261

On the integrability of the maximum and the local properties of Gaussian fields V.A. Dmitrovskii

271

On large deviation probabilities for the maximum likelihood estimators K. Dzhaparidze and E. Valkeila

285

Functional integrals, a variational method and some problems of stochastical physics G.V. ESmov

293

Contenu

v

The law of the iterated logarithm and order statistics

V.A. Egorov

304

Martingales in non-life insurance

P. Embrechts

314

Asymptotic minimaxity of usual goodness of fit tests M.S. Ermakcrv Les fonctions aléatoires à valeurs dans les espaces lusiniens et leurs modifications régulières X. Fernique

323

332

Critical branching caused by interaction with point catalysts K. Fleischmann

350

Statistical analysis and dating of the observations on which Ptolomy's "Almagest" star catalogue is based A.T. Fomenko, V.V. Kalashnikov and G.V. Nosovskii

360

Global asymptotics of solutions of the first and second boundary value problems for Kolmogorov equation with small diffusion S.M. Frolovichev

375

Hydrodynamic limit for Ginzburg—Landau type continuum model T. Funaki

382

Almost everywhere limit convergence of subsequences of powers V.F. Gaposhkin and J.M. Rosenblatt

391

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On shot noise processes with long range dependence L. Giraitis and D. Surgailis

401

G-estimates of eigen-values of covaxiance matrices V.L. Girko

409

On large deviations for trimmed sums of independent random variables V.V. Godovan'chuk and V.V. Vinogradov

424

Random operator functions defined by stochastic differential equations N.Yu. Goncharuk and Yu.L. Daletsky

433

Hellinger integrals and Hellinger processes for solutions of martingale problems B. Grigelionis

446

On connection between the theory of branching processes and queueing theory S.A. Grishechkin

455

Estimation of MANOVA eigenvalues A.K. Gupta

463

vi

Contents

On joint distributions of random elements with given mutual conditional distributions B.M. Gurevich

470

Improvement and generalization of Cramer—Rao inequality for a filtered space A.A. Gushchin

480

Limit theorems for record times A. Gut

490

On statistics of continuous Markov processes: s e m i - Markov approach B.P. Harlamov

504

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Quantum stochastic flows R.L. Hudson

512

Convergence, of the stochastic quantisation method I. A. Ignatiuk, V.A. Malyshev and V. Sidoravicius

526

Adaptive tests for minimax testing of nonparametic hypotheses Yu.I. Ingster

539

A survey of the metric theory of continued fractions, fifty years after Doeblin's 1940 paper M. Iosifesku

550

Boundary and entropy of random walks in random environment V.A. Kaimanovich

573

Non-linear law of large numbers and Gaussian fluctuation for s t a r like networks M.Y. Kelbert and Y.M. Suhov

580

Asymptotic solution of inverse Kolmogorov equation for diffusion processes with small diffusion V.M. Khametov

590

Asymptotic robustness of discriminant procedures for dependent and non-homogeneous observations Yu. Kharin and A. Medvedev

602

Characterization of distributions: problems, methods, applications L.B. Klebanov and A.A. Zinger

611

The entropy order of operators and limit theorems in Banach spaces V.I. Kolchinsky

618

Inverse problem for potentials of measures in Banach spaces A.L. Koidobski

627

Investigation of the convergence of functions of sample means and covariances to limit distributions T. Kollo

638

PREFACE T h e traditional Vilnius Conference on Probability Theory and Mathematical Statistics was held for the fifth time in 1989, June 26 - July 1. It was organized, as usual, by the Institute of Mathematics and Cybernetics of the Lithuanian Academy of Sciences, the Steklov Mathematical Institute of the USSR Academy of Sciences and the Vilnius University. T h e Organizing Committee was headed by the cochairmen Professor Yu.V. Prohorov (Moscow) and Professor V. Statulevicius (Vilnius). There were 744 participants at the Conference from 28 countries. The main directions of discussions covered asymptotic methods in probability and mathematical statistics, theory of Markov processes, stochastic analysis, martingales, stochastic physics and random fields. Q u a n t u m probability and computer statistics were included in the program for the first time. Two plenary lectures delivered by Professor A.A. Borovkov (USSR) and Professor K.R. Parthasarathy (India), as well as 446 talks, divided into 63 sectional sessions, were presented. Poster session included 172 communications. T h e present Proceedings in two volumes contain the papers received by the Organizing Committee b o t h from the invited speakers and authors of some selected short communications. We wish to thank most cordially all contributors to the success of the Vth Vilnius Conference and express hopes that these Proceedings will stimulate further progress in probability and mathematical statistics.

Copyright © 2020. Walter de Gruyter GmbH. All rights reserved.

B. Grigelionis, Yu.V. Prohorov, V.V. Sazonov, V. Statulevicius

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Piob. Theory and Math. Stat., Vol. 1, pp. 1 - 2 7 B.Grigelion» tt al. (Eds.) 1990 VSP/Moksla*

THE LOW DENSITY LIMIT AND THE QUANTUM POISSON PROCESS L. ACCARDI and LU YUN GANG* Centro Matematico V.Volterra, Dipartimento di Matematica, Università' di Roma II, Italy ABSTRACT We prove that, in the low density limit (z2 = fugacity —» 0), the family of processes given by the collective Weyl operators and the collective coherent vectors (cf. a free Bose gas), converge to the Fock quantum Brownian motion over L 2(R, dt, K ) , where K is an appropriate Hilbert space (cf. Section 1). Moreover we prove that if we couple a nonrelativistic quantum system with a quadratic interaction (cf. formula (1.10) below) to a free Boson gas in the Fock state, then the matrix elements of the wave operator of the system at time t / z 2 in the collective coherent vectors converge to the matrix elements, in suitable coherent vectors of the quantum Brownian motion process, of a unitary Markovian cocycle satisfying a quantum stochastic differential equation ruled by some pure number process (i.e., no quantum diffusion part and only the quantum analogue of the purely discontinuous, or jump, processes). The explicit form of the equation is determined and the unitarity is proved.

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1. I N T R O D U C T I O N In the last two years our understanding of the weak coupling limit has drastically increased since it has been possible to acquire control of the limit not only of the reduced dynamics, but of the full quantum dynamics, both in the Schròdinger and in the Heisenberg form (Accardi et al., 1989, 1990a, 1990b; Accardi and Lu Yun Gang, 1989a, 1989b). The basic new idea in these papers was the introduction of some classes of "collective states" (collective coherent states or collective number states) and the study of the limit of the matrix elements of the Schrodinger as well as of the Heisenberg dynamics with respect to these collective states. Both in the classical and quantum case the results available in the low density case are much poorer than the corresponding ones for the weak coupling case: for example the existence of the Markovian semi-group has been proved only for a short time interval in a very special Fermion model (Diimcke, 1989a, 1989b). Let H0 and Hi be complex separable Hilbert spaces interpreted respectively as the system Hilbert space and the one particle reservoir Hilbert space. Let W { H

X

)

=

{ W ( f ) :

f

€

H\}

* On leave of absence from Beijing Normal University © L . A c c a r d i and Lu Yun Gang. 1990

(1.1)

2

The Low Density

Limit

and the Quantum

Poisaon

Process

be the Weyl C*-algebra on H\\ let i f be a self-adjoint bounded below operator on Hi and z, /3 positive real numbers interpreted respectively as density of the reservoir particles and inverse temperature. Define Qz

:= (l + z2e~PH)

( l — z2e~/3H)~1

= coth{0H + p)

(1.2)

(with z2 = e*1) and suppose that, for each z in an interval [0, Z], Qz is a selfadjoint operator on a domain D, independent on z. Denote y'Q, the mean zero gauge invariant quasi-free state on W(H\) with covariance operator Q 2 , i.e. V Q . m f ) ) = exp and let

(/,«./>) ,

be the GNS - triple of (W))*Q.)

We Q, {Ji, the

(1.3) so that

= 0,

StQz = QzSt,

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V/ G H 1

(1.5)

where the equality is meaxit on D. This implies that the second quantization of St, denoted W(S