Probability Theory and Mathematical Statistics: Vol. 2 [Reprint 2020 ed.] 9783112313985, 9783112302712

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PROBABILITY THEORY AND MATHEMATICAL STATISTICS

Volume II PROBABILITY THEORY AND MATHEMATICAL STATISTICS Proceedings of the Fourth Vilnius Conference

Vilnius, USSR, 24-29 June 1985

Edited by

Yu. V. Prohorov, V. A. Statulevicius, V. V. Sazonov and B. Grigelonis

WWNU SCIENCE PRESSICI

Utrecht, The Netherlands

V N U Science Press B V P . O . B o x 2093 3500 G B Utrecht T h e Netherlands

© 1987 V N U Science Press B V First published 1987 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.

CIP-DATA K O N I N K L I J K E B I B L I O T H E E K , D E N H A A G Probability theory and mathematical statistics: proceedings of the Fourth Vilnius Conference: Vilnius, USSR, 24—29 June 1985/ed. by Yu.V. Prohorov . . . [et al.] — Utrecht: VNU Science Press. — 111. ISBN 90-6764-068-9 (vol. II) ISBN 90-6764-069-7 (set) SISO 517 U D C 519.2(063) Subject headings: probability theory/mathematical statistics.

Printed in G r e a t Britain by J. W. A r r o w s m i t h Ltd., Bristol BS3 2 N T

CONTENTS Preface Mixing properties for/-expansions M. losifescu Non-linear time-domain analysis of Gaussian processes Z.A. Ivkovic Malliavin calculus for discontinuous processes J. Jacod Accompanying laws for processes with independent increments A. Jakubowski Diffusion approximation of multitype branching processes A.Joffe Characterization of queues and its stability estimates V. V. Kalashnikov and S. T. Rachev On the entropy of expansions with odd partial quotients S. Kalpazidou On the rate of convergence in Maruyama's invariance principle S. Kanagawa On a general growth model possessing a subexponential growth rate G. Kersting Stochastic partial differential equations as stochastic space-time models P. Kotelenez Levy-Baxter type theorems and strong sub-Gaussian random processes Yu. V. Kozacenko and V. V. Buldygin Principal component analysis under correlated multivariate regression equations model P.R. Krishnaiah and S. Sarkar Approximation of statistics distribution by convolutions of generalized Poisson measures J. Kruopis

vi

Contents

Some general results in control theory N. V. Krylov

129

On the rate of of distributions of semimartingales K. Kubilius andconvergence R. Mikulevicius

139

On Ito's excursion law, local times and spectral measures for quasidiffusions U. Kuchler

161

Optimum design of general intraclass regression experiments and general analysis of covariance experiments V. G. Kurotschka

167

Limit theorems for functionals of geometric type of homogeneous isotropic random fields N.N. Leonenko

173

On bounds B. Ya. Levit for the minimax risk

203

Asymptotical properties of the local density of measures for semimartingales and some of their applications Yu.N. Lin'kov

211

Generalizations A.I. Martikainen of the law of the iterated logarithm

235

Limit theorems for sums of random variables with a stable limit law J. Mijnheer

253

Estimates and asymptotic expansion of the remainder term in the CLT for randomized decomposable statistics Sh. A. Mirahmedov

263

Multiparametric G.M. Molchan Brownian motion on symmetric spaces

275

Markov maps in noncommutative probability theory and mathematical statistics E.A. Morozova and N.N. Cencov

287

Classes of limit laws for functions of some statistical estimates R. Mukhamedkhanova

311

Topological and probabilistic characterizations of some classes of Banach spaces and operators D.H. Mushtari

327

Contents

vn

Asymptotic expansions of the distributions of sums of i. i. d. Hilbert space valued random variables S. V. Nagaev and V. I. Chebotarev

357

Limit theorems for order statistics based on sums of random variables V. B. Nevzorov

365

Markov additive processes: large deviations for the continuous time case P. Ney and E. Nummelin

377

Hodges-Lehmann efficiency of nonparametric tests Ya. Yu. Nikitin

391

An operator approach in limit theorems for sums of the type n~ "p S„ in Banach spaces R. Norvaisa, V.J. Paulauskas and A. Rackauskas

409

Pointed priors and asymptotics of the a posteriori risk A.A. Novikovandl.N. Volodin

421

Stokes-Boussinesq-Langevin equation and fluctuation-dissipation theorem Y. Okabe

431

The barycenter concept of a set of probability measures as a tool in statistical decision A. Perez

437

Solutions of Bogolyubov equations for infinite three-dimensional systems of particles D. Ya. Petrina, V. I. Gerasimenko and P. V. Malyshev

451

Inequalities for the maximum of partial sums of random variables and the law of the iterated logarithm V. V. Petrov

461

A characterization of a Gaussian vector based on Kagan-Linnik-Rao theorem A. Plucinska

467

Extreme functionals in the space of probability measures S. T. Rachev

471

viii

Contents

A N O D I V : generalization of A N O VA through entropy and cross entropy functions C. Radhakrishna Rao

477

Limit theorems involving restricted convergence and the continuation theory of distribution H.-J. Rossberg

495

On approximation accuracy for distribution functions of the sum of independent random variables using infinitely divisible distributions L. V. Rozovsky

501

On the kinematic dynamo problem in a random flow B.L. Rozovskii

509

On the probability of large excursion of a nonstationary Gaussian process R.Rudzkis

517

Weak convergence of integral type functionals Z. Rychlikandl. Szyszkowski

525

Characterizing the distributions of the random vectors X,, X 2 , X 3 by the distribution of the statistic (X t —X3, X 2 - X 3 ) Z. Sasvari and W. Wolf

535

On large deviations for the probability density of sums of independent random variables L. Saulis

541

On normal approximation in Hilbert space V. V. Sazonov, V. V. Ulyanov and B. Z. Zalesskii

561

Almost-even number-theoretical functions W. Schwarz Limit theorems for randomly indexed sums in a separable Banach space G. Siegel

589

Some properties and applications of Feynman measures in the phase space O. G. Smolyanov and E. T. Shavgulidze

595

581

Contents

ix

On limit theorems for multilinear forms V.A. Statulevicius

609

Characteristic functional and cylindrical measures in DS-groups V. I. Tarieladze

625

On Pareto-type distributions J.L. Teugels

649

Renewal method in the theory of semi-Markov processes on arbitrary spaces J. Tomkd

653

Innovation problem for a class of Ito processes: filtration problem for multi-dimensional diffusion type processes T.A. Toronjadze

659

On Gaussian approximation in Hilbert space, non-identically distributed random variables case V. V. Ulyanov

671

Statistics on the set of the naturals' partitions, limit forms of young random diagrams and asymptotic distributions on the space of positive series A.M. Vershik

683

Bellman inequalities for Markov decision drift processes A.A. Yushkevich

695

Coexistence of low temperature continuous spin Gibbs states M. Zahradnik

703

PREFACE T h e International Vilnius Conferences on Probability Theory and Mathematical Statistics have been held every four years since 1973 and have become traditional. However, their proceedings were never published. Only a limited number of copies of the abstracts of contributions were available, published locally. T h e present publication makes it possible for a wide international scientific community to become familiar with the invited lectures delivered at the 4th Vilnius Conference which was held from 2 4 - 2 9 J u n e , 1985. T h e main topics of the conference were (controlled) random processes and fields, limit theorems of probability theory, and asymptotic methods of statistics. T h e organizing committee of the conference invited all the 45- and 30-minute speakers to present a paper to the Proceedings. W e included in the Proceedings all of the papers received. T h e r e were 453 participants at the conference of whom 150 were invited speakers. W e hope that the 4th Vilnius Conference and its Proceedings will give rise to many important and interesting investigations in the future. Y u . Prohorov

xi

Prob. Theory and Math. Stat., Vol.2, pp. 1-8 Prohorov et al. (eds) 1986 VNU Science Press

MIXING PROPERTIES FOR/-EXPANSIONS Marius Iosifescu Centre of Mathematical

Statistics,

str. Stirbei

Vodâ 174, 77104 Bucharest,

Romania

Abstract—Several c o n d i t i o n s o c c u r r i n g in t h e s t u d y of piecewise m o n o t o n i e t r a n s f o r m a tions a r e discussed. U s i n g the F r o b e n i u s - P e r r o n o p e r a t o r c o n d i t i o n s a r e given u n d e r which either reversed (¿»-mixing o r ^ - m i x i n g hold for / - e x p a n s i o n s .

1. P I E C E W I S E M O N O T O N I C T R A N S F O R M A T I O N S Let T: U-*I= [0, 1] be a continuous m a p , where C / c / i s open and A(£/) = 1 (¿ = Lebesgue measure). Then there exists a finite or countable collection (I„)asA of closed intervals with disjoint interiors such that l j a e / , laz> U, and for any aeA the set Iar\([\U)consists exactly of the endpoints of Ia. Assume that for any aeA the restriction of T to / f l n U is strictly monotonic and extends to a C 1 -function Ta on la with TJIJ = I. A m a p 7" with the above p r o p e r t i e s is c a l l e d a C1

piecewise

monotonic

transformation.

For any a e A let fa denote the function inverse to Ta, thus mapping I onto Ja. F o r any aM = (al,

. . . , a„)eA",

put

/.«=/.; ' ' ' °fa„ (o denotes composition of functions). Clearly ,/„ 0, J 7 r dX = 1 and Pr = r, i.e. p is ^-invariant. If p' with dp' = r' dX is ^-invariant, then r' e BV(I) and p' is absolutely continuous with respect to p. Assume from now on that condition (C) holds. [Remember that, on account of Proposition 3, (C) implies (E m ) for some m.] It is well known (Renyi, 1957; Halfant, 1977) that under (C) there exists a ^-invariant, ergodic probability measure p which is equivalent to X. The density r = dp/dX of p satisfies almost everywhere on /the inequalities 1/C^r^C.

(3)

4

Mixing properties for /-expansions

Moreover, as shown by Rohlin (1961), the dynamical system (I,38, p, T) is exact. Then Theorem 4 implies uniqueness of a probability density h (with respect to p satisfying the equation P-ph = h, the only one thus being h = 1. As to P-p it is easily seen that Pr h =

r

\

heLl(I,

3), p).

[Clearly, on account of (3), the elements of the spaces Ll(I, 38, p) and Ll(I, 38, 1) are identical. We shall simply write 11 - J11 for the norm in any of these spaces whenever confusion is not possible.] Finally, an easy computation shows that assuming in addition condition (BV), the operator Ps satisfies inequality (2) for h e B Vp (I) as P does for h e B V(I), with suitable k, d and D. [The definition of BV-JJ) on analogy of that of BV(I) = BV>(I) is obvious. The remark above concerning Ll(I, p) and Ll{I, 38, X) applies to BVJ1) and BV(I) too.] Thus P-p enjoys properties (i) through (iii) of P. It appears that p is in fact p defined above, and r = reBV(I). Also, s = l and ). l = 1 is the only eigenvalue (of multiplicity 1) of modulus 1 of both P and P. We can therefore state Theorem 5. Assume conditions (C) and (BV) hold. Then for any h e BVp([) and n ^ 1 we have IIP"h -

h dp|| 0 and 0 < q < 1. The proof follows from the inequalities l|, ^ess sup |/i(x)|


for suitable positive constants c, and c 2 not depending on /"". But X

p(/.«) = p(7)= 1,

/c>l,

and sup

£

var /a'(k> < oo

on account of Proposition 1 of Rychlik (1983, p. 73) (take there/= 1). Finally, noting that p({rk} oo to the standard Brownian motion process.

M. Iosifescu

7

4. S O M E C O N C L U D I N G R E M A R K S As has already been mentioned, Renyi's condition (C) was originally introduced for /-expansions. Subsequently, it was dealt with for piecewise monotonic transformations (Adler, 1979; Halfant, 1977). The restrictive assumption Ta{Ia) = / , aeA, is precisely reminiscent of/-expansions, where it is quite natural. In fact, it is not at all essential and one can instead assume that inf

A(TJ/a))>0

aeA

which always holds when A is a finite set. Without such an assumption there can be no absolutely continuous T-invariant measure (cf. Bugiel, 1985). It is certainly interesting to note that, on account of Proposition 1 here, the assumptions in Theorems 1 - 3 of Lasota and Yorke (1973) imply Renyi's condition (C). Thus these theorems d o not improve Renyi's result concerning the existence of a Tinvariant density, save the additional information that this density is a function of bounded variation. The great merit of Lasota and Yorke's paper lies in the new approach it has initiated. The relevance of Ionescu Tulcea and Marinescu's ergodic theorem in this context has been first noted by Keller (1979). This theorem is motivated by and in turn is of fundamental importance in the theory of dependence with complete connections. F o r piecewise monotonic transformations conditions (C) and (BV) ensure the absolute regularity of the label sequence ( a J •))„;>,. Under just (E J for some m and (BV) the only thing which can be asserted is the existence of a natural number k such that (akJ is absolutely regular [cf. Hofbauer and Keller (1982, p. 128)]. It would certainly be interesting to know how (C) and (BV) should be strengthened in this general setting for to get for the label sequence the types of mixing occurring in Theorem 6. Finally, Theorem 6 was stated under stronger assumptions by G o r d i n (1968), who gave no proof. I d o not know of any other instance in which reversed Oy

Suppose

that

in

Ivkovic

^ " H f )

is 0

in

i

F ( t ) = inct)||2 = ) l V t ) H 2 + l h d ( t ) l f = F c Ct) t Fd Ct) the discrete

part

i,J | l M f wherej T T is the spaee of laws of processes with independent increments /PII/ and trajectories belonging to the Skorokhod space D ( R + : R 1 ) ; it is additionally assumed that if k ( k ) .

will denote the scalar product of the vector

Theorem I.

Let

Z ^

x

and

y .

be a sequence of multitype Galton-Watson

processes such that (i)

= I + 1 C n

(ii) lim

= c, ,

n

I = 1

with

lim C n-*»

d ,

= C

ct^ s K

for some

1 J

n+co

a

K ,

i == l,...,d , (iii) lim I (r.-mj n J) 2 p| n ) (r) = 0 , for any L n r:||r||>e /n 1 ^ i = l,... d

e>0

Then, given a sequence 3 ^

the

e

such that

lim

X n (0) = x < n )

,

n

ne

n

= a , a > 0

processes X< n >(t) = e / ^ C n t ]

,

lim x< n > = x Q , x q * 0

will converge weakly to the unique diffusion2 process in R + d starting at Xq with generator given on C functions by: d

L :=

I i=l

(xC)

1

a X

i

d l 1« ^ * 1= 1

X'a,

a2 3x£

.

A. Theorem II.

Let

Zn

33

Joffe

be a multi type critical positive regular

Galton-Watson process with mean matrix ck j .

Let

R

and covariance matrices

be the projection matrix in the Perron-Frobenius

decomposition of / \ / \ X^ '(0) = XQ '

M

M .

Then if

X^(t)

converging to

= e z" , n LntJ

XQ * 0 , ne n

with

a , a > 0 ,

the

sequence ?(n)(t)

=

x(n)(t)

.

n

f

x(n)(x-)(M-I)dAn(T)

n Jo will converge weakly to the unique diffusion in the half space

xR a 0

with diffusion operator

L

given by

Lcp = aXR tt Y a. . ~ — ( a l l 2 .. ij 8x i 8x j and initial condition

3.

tp C -functions on

xR a 0)

Xq .

CONTINUOUS TIME

The model.

We consider a system of

d

types of particles.

denotes the rate of death of a particle type of type P.j(j) = P q -(j 1 -• • .i d ) particles of type Z(t) = ( Z ^ ( t )

i

the probability that when it dies

l,...,d

are created.

Z^(t))

number of particles of type

A.

and j-j

jd

Let

denote the random vector of the l,...,d

in the system at time

t .

We use the following notation

V.£(i)

=

I

oj » = (a „(l)...a „(d)) , in yAi iiiji, m $ JL

' al =

where

a

T

t,lW

denotes the transpose

Multitype branching processes

34

e^ = (6-j j,... ,6 d j)

the

jth

unit vector

a = A(M-I) N+d

the set of

d-dimensional vectors whose components are non-

negative integers. Under those assumptions it can be shown that this system defines a unique Markov process

Z(t)

We consider now a sequence

whose state space is (Z^(-))n>Q

the above notations with superscript n

n

X< )(t) = e n Z^ ^(nt)

of such procseses using

n .

We introduce the sequence

n

(noting that the

.

Z (nt)

are as

Z n (t)

but

with an increasing intensity of jumps which come to the same as accelerating the time). We shall modify the model to deal with population-size dependent multitype branching processes, generalizing the results of Lipow (n) (1977).

Keeping the above notation, we let the quantities

and

p^ '

x e N+ci .

be functions of

We make the following assumptions: (i)

sup sup A ^ ^ e ^ x ) n x

(ii)

(iii)

< °° , lim A ^ n+°°

sup sup n a n ( e ^ x ) < °° x n

= A(x) ;

lim na ( e ^ x ) = C(x) ; n-t-°°

lim sup sup I ||j||2p(n)(j,x) = 0 , I = l,...,d , N-t- n x j:|j|>N

(iv) sup sup a j n ) < - , n x

lim \ [ n ) n+°°

*) = a £ (x) ;

2 (v) Let on

CQ R1^

be the space of and

L

C

be defined on

bounded continuous functions Cn

by

A.

L =

I i=l

(xC(x)). 1

dX

i

35

Joffe

+

§

e

1 xfyix) 3x 1=1 1 1

Then the martingale problem associated with

, a > 0 .

(L,C q ,XQ) , XQ e

has a unique solution (this i s true in particular i f C(x)

and

a(x)

Theorem I I I .

d = 1

or i f

are constants).

Let

Z^(t)

be a sequence of population-size-

dependent multitype branching processes, s a t i s f y i n g the assumptions ( i ) to (v), above, and let

X^(t) = e Z^int)

with

lim c n = a , a > 0 and X ^ ( 0 ) = xi n ^ with lim x j ^ = x n , n / \ u u u x n * 0 . Then the sequence X^ J ( t ) converges weakly to the +d unique diffusion in R starting at xQ with generator L . Theorem IV.

Let

Z(t)

be a c r i t i c a l positive regular multitype

branching process, with parameters

a = A(M-I)

and

o .

Let

R

be the projection matrix in the Perron-Frobenius decomposition of exp (at) .

X ( n ) ( t ) = e n Z(nt)

Then, i f

converging to

Xg , xQ * 0

and

with

X(n)(0)=x^n)

lim ns n = a , a > 0 ,

sequence of martingales

the

t

/(n), X V " ' { T )adx 0 will converge weakly to the unique diffusion in the half space (4.4.10)

xR g 0 ,

?(n)(t) = X(n)(t) - n

with generator

L

given by

Lcp = aXRA I and i n i t i a l condition

XQ

(