Stability Problems for Stochastic Models: Proceedings of the 6th International Seminar Held in Moscow, USSR, April 1982 (Lecture Notes in Mathematics, 982) 3540122788, 9783540122784

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continuation on page 301

Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann

982 Stability Problems for Stochastic Models Proceedings of the Held in Moscow, USSR, April 1982

Edited by V. V. Kalashnikov and V. M. Zolotarev

Spri nger-Verlag Berlin Heidelberg New York Tokyo 1983

Editors

V.V. Kalashnikov Institute for Systems Studies Ryleyev st, 29,119034 Moscow, USSR V.M. Zolotarev Steklov Mathematical Institute Academy of Sciences of the USSR Vavilov st. 42, 117333 Moscow, USSR

AMS Subject Classifications (1980): 60 B 10, 60 B 99, 60 E10, 60 E 99, 60F05, 60K25, 60K99, 62El0, 62FlO, 62F35, 62H12, 62P99 ISBN 3-540-12278-8 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12278-8 Springer-Verlag New York Heidelberg Berlin Tokyo

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.

© by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210

Con t e n t s Zolotarev V.M. Foreword,

VII

Anichkin S.A. Hypererlang approximation of probability distributions on (Q, 00) and its application • . . . . . . . . . . . . . . . . . . Azlarov T.A., Volodin N.A. On the discrete analog of Marshall-Olkin 's distribution

17

Gnedenko B.V. On some stability theorems • . . . . . . . . . . . . . . . . . . . . . . . . . 24 Hohlov Yu.S. On stability estimation of certain characterization of the exponential distribution

32

Ivnitsky V.A. Accuracy estimation of the results of complex systems simulation with vector output and several types of randomnesses..................................... 38 Kalashnikov V.V. A complete metric in the function space 1J (OJ 00) and its application.................................... 60 Klebanov L.B., Manija G.M., Melamed J.A. On the estimation of location and scale parameters of stable laws

77

Klebanov L.B., Melamed J.A. Discretization in the problems of stability of characterization of the exponential distribution • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 M.V., L.B. Some questions of stability theory of the stochastic economical models . . . . . . . . • . . . . . . . . . . . . . . . 123 Obretenov A., Rachev S. Characterizations of the bivariate exponential distribution and Marshall-Olkin distribution and stability,. . . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . 136 Ostrovskii I.V. On the growth of entire characteristic functions .'

151

Panaretos J. An elementary characterization of the multinomial and the multivariate hypergeometric distributions ,

156

Petrova L.L. On the stability of characterizations of the unit distribution

165

Rachev S.T. Minimal metrics in the real random variables space

172

Rydel A. On poisson output of queueing systems

191

Senatov V.V. On a relation between Levy-Prohorov metrics and ideal metrics • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Shimizu R. On the stability of lack of memory characterization of the exponential

217

IV

v

Siganov I.S. Several remarks on applications of one approach to studies of characterization problems of Polya I s theorem type . . . . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . . . . 22 7 Szulga A. On the metrics of the type

238

Wolinska A. On a problem of DUguJ

24 4

Yanushkevichius R.V., Yanushkevichiene O.L. Limit theorems in the problems of stability

2 54

Zieli6ski R. Robust statistical procedures: a general approach. • • • . . . . . . . . . . • • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 83

Reports not included in this volume Anichkin S.A. (Moscow, USSR). On coupling of renewal processes and its application: a review. Bednarski T. (Wroclaw, Poland). On minimax testing between Prohorov neighbourboods of probability measures. Bentkus B., Rachkauskas A. central limit theorem in

(Vilnius, USSR). Rate of convergence in the spaces in different metrics.

v

Cerkasov I.D. (Saratov, USSR). On optimal deviation of the diffusion process from the stable deterministic chasing.

"C istjakov

G.P. (Harkov, USSR). On stability of decomposition of infinitely-divisible distributions.

Dimitrov B. (Sofia, Bulgaria), Klebanov L.B. (Leningrad, USSR), Rachev S. (Sofia, Bulgaria). On stability of some characterization of the exponential distribution. A.A. ic models.

(Novokuznetsk, USSR). The problem of robustness of

v " Ginzburg B.l., Sarsanova G.P. of characteristic functions. HUu Nhu,

Ngok

(Harkov, USSR). Analytical peculiarities (Hanoi, Vietnam)

e

-sufficient statistics.

Illin V.V. (Leningrad, USSR). On the estimate of the stability of decomposition of the normal law on the components. Klebanov L.B. (Leningrad, USSR), Mkrtcjan S.T. (Erevan, USSR). Some problems in stability of characterizations of distributions. Landsman Z.M. USSR) On some classes of the families of distributions, characterising by the minimal Fisher information property. Lisek B. (Berlin, DDR). Some new results on the existence and stability of the solutions of the equalities of the type- Y = S' (Yo, X,j ) nt 1

v

Malosevskii S.G. (Leningrad, USSR). The estimate of stability of the characterizations of distributions with the help of some first moments.

"

Muhin A.B. (Taskent, USSR). On the stability of characterization of the lattice distributions and its application. Rachev S. (Sofia, Bulgaria). On metrization of weak convergence together with the convergence of the moments. Seidl L. Hungary). On the continuity of the certain class of the networks of gueues.

"

Seryh A.P., Tarasenko F.P., Sulenin V.P. (Tomsk, USSR). On the asymptotic properties of the family of the stable R - estimates of the 01. location parameter.

VI

Skrjabin D.O. (Jakutsk, USSR). On the stability in the problem of reconstruction of the distribution by the distributions of some statistics. v

Surygin A.M. (Moscow, USSR). Robustness and optimization of the asimptotic stability of the estimates of parameters. v

Tsitsiashvili G.S. (Vladivostok, USSR). Probability metrics and orderings invariant to transformations giving by the signs of derivatives. Tsitsiashvili G.S. On the problem of the statistic of mixing flow.

fl

V.E. USSR). Many-dimensional -stable distributions and realizations of generalised convolutions. Vyndev D., Ignatov Z., Rachev S. (Sofia, Bulgaria). On the metrics invariant to monotone transformations. Weron A.

(Wroclaw, Poland). On stable measures in Banach space.

Yanushkevichiene O.L. (Vilnius, USSR). On the stability of characterization by the lack of memory properties in two points. Ja. (Praga, Czechoslovakia). Asymptotic behaviour of -estimators of location in nonregular case.

M

Zinger A.A. (Leningrad, USSR). On the generalization of the characterization theorem due to Linnik-Anosov and its stability. Zinger A.A. (Leningrad, USSR), Yanushkevichius R.V. (Vilnius, USSR) On the stability of characterization theorem due to Yu.V.Linnik. Zolotarev V.M. (Moscow, USSR). On the method of metrical distances in stability problems.

For e w 0 r d The place in

__frorn

of Stochastic f)tCi1:)i}i!=-X took Al?ril 1982. The previous seminars were

organized as follows: in Leningrad and Vilnius (1974), in Moscow (1977), in Palanga, Lithuanian SSR (1979), and in paneveiis, Lithuanian SSR (1980). The

material

of these seminars

was

published in four

special proceedings (1) - [4J . Short information about the 4th and 5th seminars and summaries of some of the papers were published in

[5J

and [6] • The 6 th seminar was organized by the Steklov Mathematical Institute of the Academy of Sciences of USSR (MIAN), the International Research Institute of Control Problems (MNIIPU), which assumed the greatest part of the care of organizing the seminar, and the Institute for Systems studies (VNIISI). The organizing committee consisted of Academician Prof.S.V.Emeljanov - Chairman (MNIIPU), Prof. V.M.Zolotarev - Vice-chairman (MIAN), Prof. V.V.Kalashnikov (VNIISI) and Dr. V.I.Lutkov - Secretary (MNIIPU). There were 47 reports submitted by representatives of well known mathematical centres of seven countries. Those reports not included in this vOlume are listed by title. The present proceedings include papers based on the reports submitted at the seminar, as well as a paper by the Japanese mathematician R. Shimizu (presented at the Soviet-Japanese symposium on Probability theory, held at the end of August 1982 in Tbilisi), and a paper by the Greek mathematician J. Panaretos (delivered at the International Conference on Probability theory, held in Vilnius in June 1981). A characteristic fe2ture of all six seminars on Problems of Stochastic models stability was the great variety of topics in the rapers delivered. Though most of the reports have always been devoted to the main topic of the seminar, the papers presented here featured such problems as characterization of distributions, investigations of probability metrics, characteristic functions, etc. The point is that the respective interests within probability theory and its application of most of the participants are often qUite far apart from each other. The fact that the seminar managed, during its existence, not only to retain, but also increase the number of "permanent" participants can be explained first and foremost by a tradition, created at the seminar, according

VIII

to which every participant could pose any problem in which he had taken a lively interest. The discussion of the contributions to our seminars was performed from the viewpoint of their general tendency, i.e. from one of the general conceptions of stability. Of course the participants interest in the stability problems isn't of itself sufficient for such a practice to be a success. It is necessary to have also some general conceptual and methodological approach to the stabi lity phenomenon. During the existence of the seminar such a general approach was outlined and here we give a brief review of it. The notion of stochastic model stability can be explained within

f J

the framework of the following scheme. Let !Je = {X} and dfj ­ Y be sets of random variables defined on a probability space and taking values from the corresponding measurable ­spaces

(V. !J.ry )

. Suppose that some mapping

is given and the sets

}/;

X

tions imposed on the random images in

djj

and

IF'

(

1J,

of the set

1b g "!:!

5t

and into

(see Fig.1).

$r ={X}

Y={Y}

X6 ­ ­ ­ ­ . _ ­ ' ­

­::>'7.

3e' Fig. 1 The problems connected with the triplet (JfJJ§,f) are divided into direct problems in which one is interested in the properties of the values from the set

(! = ..5'5 (I

f ::It

,

and the inverse ones

in which the object of the investigation is the set (here denotes the pre­image of the set

IF

'!J

describe the condi­ tl't' and their variables mapping from

IX

Note that two important particular cases of this model corres-

.:II: = !1C

pond to large classes of problems when

or

CY.

=

The concept of stability of direct and inverse problems is rela ted

to

certain

the

and

probability metrics

11

t'

V JlI £: 1£ .

and

and to a certain set

defined on

Under the notion of the probability metric in

06.(-1

any functional

(X,

XI)

we mean

defined on the set of

00

)(, 5(1

joint distributions of all pairs of the random variables from

and having the pseudometrics properties in

, i.e. the

symmetry property, the triangle inequality and the property

P(X- X') = 1

lity theory are, like

t>

f/(X,X I ) =0,

All distances used in probabiobviously, probability metrics, however distances

I E(X­ XI) I

also belong to them (if

V

is a Banach space).

If we consider

as the usual distance between an element and a set we can say that the direct problem, connected with the triplet stable with respect to the

e = f (X: It) + V (fFt " JtJ) ­ implies

( .It,

the set

ftJ, If)

is if the relation

0

tJ'=Y(fX;C)---O,

10 nf 1l

where (! = The implication corresponds to the notion of a qualitative stability of the model, while any estimate

tf ep (c)

,

where 'I'(£)--+-

0

»

when

C'-o

is a solution of a quantitative stability problem of the same model. Such a comprehension of the mathematical model stability is very capacious and is able to include practically any problem of approximating type. For instance, it includes the general notion of robustness, proposed by Humpel

[71

and the notion of differential equation stabili-

ty (according to Liapunov) with random perturbations, developed in the works of Kushner [a] , etc. It turns out that in contrast to the quantitative stability, the qualitative stability depends very little on the structure of the mo­ del.

It can be treated as a consequence of such general properties

like the closeness of the sets the mapping

F

and

jB

,

the continuity of

and the relative compactness of the set

respect to the implemented metrics (see

[9J. ).

jtl

with

This important fact is

actually the background on which the specialists in different branches

x of the probability theory could unite their efforts in investigating the stochastic model stability. At the same time, the main interest (from the view point of the applications) is in the quantitative estimates of the stability. For instance, they can show us the boundaries of applicability of the model in hand (see [10J ). Obtaining quantitative estimates of stability, especially of the correct order, is usually the far more complicated problem than the one of qualitative analysis of stability. It requires more details about the structural peculiarities of the model. The metric treatment of stability problem assumes a good knowledge of different properties of probability metrics. Though the theory of probability metrics is still at the stage of development and accumulation of facts, one can already speak about the existence of a common methodological approach to the solution of the stability problems. We call it the metric distances method. In stochastics, probability metrics have been used for a long time. However, the choice of the metrics to be used in approximation problems is determined by topological reasons, computational conveniencies or just by the established tradition. The method of metric distances is the tool for solving the quantitative stability problems, i. e. the problems of obtaining estimates

0

o.

X,

)::;

we have = and Y( fF' 0 That is why, taking into account the properties

Jf

of the vectors from the sets

where

for

and

and

we can state that for any

are random variables with distribution

and

G(X)

respectively, and

O=Y(fF'X:C)=y(UlX;U?Il) Since the vector

X=

(Xi. , ••. , X n )

=

v (fflX;X).

which consists of the

initial random variables, has the same distribution as the vector

XE $

we have

c= y(X:1' Y), The problem of estimating the exactness of the approximation of the function

H (X)

by

(}(X)

turns out to be equivalent to the problem

XIV

of quantitative estimation of the stability of property (1), i.e. a

cJ

construction of the estimation The chosen metric (see Zolotarev [4J Let .J y

(X',

x'

X"'/

C;.

and has the following properties:

X"

and

wi th the help of

belongs to the so called "ideal metrics" be such random variables that the distance

w

is finite and

is a random variable which does

not depend on them. Then 10

2

0

• •

V (X'c W,

x". -rV)

For each constant

6

V (X:

C? 0

As an elementary corollary

of property

10

and the triangle

inequality we get the inequality

y (X/, " . eX11 (4)

{, V (X i '

)

+ ...

c.

,

f

.

0

V ( X 11 j

Yn

) 6

x; ) = n Y (Xi, Y)

Thus, from (3) and (4) we obtain

This way, since the distribution variables an estimate on

H (X)

X1.' . . . i Xn of

Fex)

of the independent random

satifies condition (2) we get the exactness of the approximation of the distributi-

by the distribution

G(X)

in the metric

Y

(5)

If we want to obtain an

metric

L

estimate in terms of the

or in terms of the uniform metric

f>

'

then we have

to solve the problem of comparing these metrics with the metric Let us do it. The Levy distance between the distribution functions and

G(X)

V

H

is defined as the length of the side of the maximal square

xv which can be imbedded between the complimented graphs of the functions

F

and

G.

Whence

Thus

s-ol

L (Z111 Y ) (2 e) 1/(f+S)n- cX(HSj S

(6)

The distribution

G(X)

= c
t}

P[

k1 = k2.=o

I f we suppose that

(15)

Summarizing both

'';2 7k:d

.

(16)

in (16) then

and we obtain (5). After that we apply theorem 1. The necessity of 1)

- 3)

is evident·

It seems that the form of the discrete analog of M-O - distribution for multidimensional vector coincides with form of M-O -distribution. But problem of describing of values of parameters

A. . ti.

'e ...

.

lJc

is open. Let us describe the form of discrete M-O - distribution.

fl.

Let Hties and

OV1=

be outcomes in experiments with the associated probabi-

t

. P(If. If -) = Io, 1!::io

V(Z) =

IJ

o

In our case we have

Then

=J y-l(l-fyf isin(t bJj U+f/))cl'j l "'0

-:1m 1f(ifit) Let

and

We have

-1m ilt') D

o

u

=

t 1

'IJf(1+it)=-t

u e (0,

The function

-1

sin

and the fJ'nction

This function is positive for

u e (7t,2Jl).

J(e-U/t-J)

then

U

1i:)

exp(U/t -1.)

s1nuda.. is periodic.

and is negative for is growing for

te (0,00)

and is positive. If we divide the integral into parts, which correspond to the semiperiods of the function sin u ,then we get the alternating series. The first term of this series is positive. Such series has the positive sum. So that we have

S(e u/i:- i) ()O

-t

I.

Lemma 2. Let

-1

o (2.)

(1)

X:1' X 1.

be posi ti ve random variables and

X1.(1)have an exponential distribution,

Then

sin u du-:a . 21,(°=

&J9xtJ,

L (X (1.) X (2») 2 L (Z (1.) Z (2)) t »

1.

1.'

1.

•

i =1.,2.

'

35

F(X)

Proof. Let

X(1)

G(X) F(e") Zi (2) ·

and

X;2). Then Zf (J.) an d

and of

be the distribution functions of and If

G (eX) are the distribution L (2 (I) Z (2)) = I then for

1.'

we have

f

or

where

=

eX> 0,

0 ,

. Be­

depend on some technology parameters

which are fixed. We shall study only the case when

the output characteristics are determined by Monte­Carlo method. We shall

consider that the vector

Cl =

(01. 1 "

the simulation takes place in the only point

" , eX n

e

)

E Q

is fixed and

•

In p.2 the formulas are given for the bias and the covariance matrix of estimates

"

... , " .

These formulas are represented

as sums of component due to different types of randomnesses. Then we obtain the concrete algorithms for different types of randomnesses.

40 2. Randomnesses of several types. Let random variables contains

e

are divided on

classes, such as

r=1, .

contains data with randomness random variables of the

4i

-

.-

((1.,'" , l; e ),

rf-,(j)(z ji ' .

'¥

"'1

i = 1., ...

Z·) Jkj

I

p, Instead

(C;ji

? ••• ,

c:P

(:;jk.) j

'"',tJ

Let

(D

p,

We agregate

z: (J)

Zjk;)'

(Zj1. , .. "

'»

CZj iJ · · · )

tp

may be treated as

of unknown distribution

of random variables

we shall use the distribution some estimate for

e.

J

11' -st class into vector l:; = (

Zk )

be unknown and we use the random generating from the corresponding empirical

[7J ' i.e. randomness;(2

is due to the random de-

viations of empirical distributions from theoretical. We suppose that

l;1, ... ,

Pl t;i >

- (d)

t;k ) is

•••

)

d

the

lation,

II «; ... ,

M (i=

, • • •

m=oo

(;k)

-th realization of (if,""

(£-1)

-

(l:; i

components

, ... •

£h)

(namely

being generated during the simu-

C,t._l j:) '0

.

(38)

68

wx(oj= inf max (;

J

LJx(oJ

TJ)

T'?O

(39 )

It is evident that

tim wx(oj =0 if and only if

to«

$-0

Wx (d, T) = 0

Hence, we have the following statement.

XED[o/oo)

Lemma 4. I f

then

iJ:t(r})-

0

when

0- 0,

The following theorem characterizes the compact subsets in

D [0,00)

(see also Theorem 14.3 [4] ).

Theorem 2.

B

A subset

from the metric space

has a compact closure iff 1) for any

XCt)E K rr 2)

T

there exists such compact

t:!:. T

for all

J

XE

8;

KrC

(D [ 0,DQ) J

E

f)

that

lim sup cJx(o)=D.

0-0

!tEB

l:1:QQf is simular to that of Theorem 14.3 from Now let us consider the projection 1[t . 1"" t1(

is

(nere

k-fold

production

E x ...

X

E

[4] .

D[O,oo)- E

k

) which is defi-

ned by

(40)

D [0,1] , see It is easy to prove (as for k maps (40) are - measurable where (1),8 )

o-

algebra in

is the Borel Let

])

[0,00)

6" -algebra Cc [0,00) .

generated by distance in

Ek

We denote

[ 4] ) that the

f

is the Borel and

61

f

k

(57)

A

jWoJ!(t)-W'°f(t)!6

It )

(43) we have

I Ui ­ U;.I).

=1

Hence, using the definitions of

i

(44) and (46) are true and con-

andf introduced in the section 5.1. Let us

and

(I

J.c,

/ )

Z (i11;;-Vilt/U i - U'i

J .

i=1.

­Ie

(58) From (58) with the help of (12)

d

l

f:

min max{fmax (tn , n

­1

?-' (1V{-l{'1 +lUi -u; I)J. 11

1=1

(59)

73

Using (50).

(59) we find

min mOf,r{i{ 1l

11

,max/en is«

l

I

rz J

1=1

(t1Jt -V;/f-IUe-U: J)} (60)

As in the section 5.1 we receive that in the conditions (44), for any

C70

(46)

6=O(C)70 that the inV(V,V') 1

L..

y=[ 1.

'::l

(1'1/1)

uC't"

" i

?Ie).l

v

1.

)

. ('1 f.) k 2. iJc. 10)/0 ) (8)-8 C '"(8) \' \' it2.

(8 f2. (Y.1 !2. ) D.

ci i

e.;2

it

c.

i=fJl=1. 2.1,

of)

U . (fJ)

it) 2.

(61)

97 Lenuna 3.

fiiC

rv

(12.) ",

- ' ". (11.),... U1'1 = ( In (c 1.j. .1\.j -

The random vector

(12)

)'

f.",(2f),...

n Cfj lj-C tj (8)! ,j=l,... ,k)

Ifn

covariance matrix of (k) of

,)

)

iJ

m)

",(22)

j .== 1, ... , K ) J

Jf (9, !B ;k»)

is asymptotically normal

(2[)

tu)

C . (8)), ,

and also the elements of the

converge to the corresponding elements

Si

Proof.

We show first that

t:m.,..P{

(d,p, if},f;)} =0, q,= 1, ... h.

=f

J

(62)

We have

p{Uxl/t

;} =

Jt (63)

+

11I1(

Let

n1/ ,1!-k(r:i)j3,Acv ,l q, )- (8)J < e ,

En = [ CJ: I t1k

Denote

zt (0)

0 0 J

is analy·

and bounded on the real

is sufficiently large. Since

toft 1ft etJ/ =

then by the Phragmen-Lindelof

,

p.71) the function

9' 0 )

ft (t)

j3!(8ff) )

is valid in

ro (AI', /3/(/3+1)

.

Similar reasoning shows that the last estimate

tIm t 0) ),

is bounded in

wff t: I f (t) j =

Consequently

=(Oltl:r (

=

i=X, r=IX/ i J(j

in

theorem ( [3]

&i"M(1")

CP(r)

f (i) exp{- c (- ii)J3!(/Hi) }

=

C.., 0

o--{ltI)

tog M(r)

() (IX /J3!(j3t1) ),

tic in the half-plane axis when

9 I,

Put in (2)

•

!!oa+ I f (X) ) =

we obtain

tJ

and therefore

(J3+i)

Thus

arbitrary then

so

Since

e=o.

0 (r)} ,

exist the number

1"=1"_00 1D then the following relationship is true

aX

and

)

r

}

The question of stability in the problems of unit distribution characterization by the property of regression constancy was investigated by E.Lukacs

[8]

)(

X)( 1.' 2.

Theorem. Let

random variables, O(.{

Xj +

(with finite expectation) has

X

regression on random variable

L{ =

[9] , [10] .

and Yanushkevichius

Definition. Random variable

,

and suppose that

e

has

(

L (FX{

/ae./=/a)

(Ue)C afn

2.

(2)

I

e'

(3)

/.

[llJ §

Remark. You may easily convince (for example no restriction to assume that due to Yanushkevichius

.

OE(O,i)

[91 ,

e (1otJ)/3

in (2) and in (3)

[10] .

Ib1 I

In the paper by Lukacs

[8J .

Also we want to prove

(2) and (3) by any common method. It should be noted

C -regression formulated above is more com-

mon as compared to the one in paper

[8J .

Proof of theorem. Let RCL..2.)=E(Lt./L 2 ) , We obtain from these relations that

E(L{eocpCitL 2 ) = k (t ) ,

kef) =Eexp[i't:L:z}R(L.2 ) , Le.

at f '(gjt) f(62.t)+Q2. f( 61t) f f{B 2t) = if< (t) 0(

both

Thus our task consists in proving (3) and therefore

that the definition of

Denote

5.3) that there is

The estimate (2) is

is only of such an order, where

we may recieve the better result then in relationships

X2 1

then

(i+"3 e)e3 tn

L

s.x, +

f

Ig2/.(X,Y; ep)=

=max{ Sup

lie 13

a

sup inf moX'[ 8E'fJo

t(II,13), fCY,1I

X,B)]}

174

I"(II,B)

where

is the Hausdorff distance on the plane with

In accordance to ce in

x- :l

!£ . x

Zolotarev (1976)

v- (X, Y, t )

Let

[0, oo )

I\.

-distan-

satisfying the following conditions:

P(X=-Y)=1

I)

let us define the

be a nonnega ti ve function on

V'(X,Yj i ) =0,

implies

'V(X,Y;t) =V-(Y, x.i),

II)

III) i f

V-eX,Y;t) -q1!(X,Y;t"),

then

11 ( X, Y ; t '+ til )

IV)

Then the functional

All (X. Y; 'It) =

1J

ex, Z, t') + 11 (Z, Y; t ").

inf fmax[ 11 ( X, y; ). t ) 7 t] , t 7 a]

is a probability metric in

!£

./170 (Lemma 7, Zolota-

for every

rev (1976n

A ""70

Theorem 1. For every

RJt eX,Y)= 1I.]a (X, Y; V), where

V(X,Y,'i)=rnax[sup inf

/I (t)

and

t5

=

=infft70; f1E33

every

E

D

CfCX,fI;Y,5),Sup in! Cf(Y,fI;X,B)}

»)-e ,

sup

inf

BE'l30

such that

For

'P(X,I/;Y,B).

the metric

,using the fact that the metric

Hausdorff distance in ] . Let us denote by sequence

etB11.

F, Fn

H"CFn,F) --0 coine ides with

etro=,,)

i f and only i f

r CF) .

then there exists

(Hausdorff (1937)).

are distribution functions then

H" (Fn 1 F) =

Proof: I f

is a

the topological limit of the set

BncD, n=1,2,...

Theorem 4. If

H

JI (

exists and

r(Fn ) r(F)) - a

ttf(Fn)

?

etfCF'n)=f(F)

and

(Hausdorff (1937)). Now,

let

Therefore, there exist a subsequence

Suppose that

Fn ,

H). (Fn , F) 7- U.

,and a positive number

180

eE

(0,1.)

such that

r (F) = et rc Fn)

and

For everyn

I

we have two possib'

le cases:

E,n E '

I) there exists a point

(en"

such that

fCF))7e?

'7n

II) there exists a point

r (Fn, )

1

E

r (F)

,

such that

The cases I) or II)

n'

lized for infinitely many numbers

l; n"

exists a subsequence

Eim. 11/1 = Let

.N"

then

c

2/1 (e, f( FJ)

there

such that

(I:; n

/I

eD

If for some

r(F"JI)

E

(or both) are rea-

.

l;G:fCF)

But

so we have a contradiction. Let

(611 1, Un')E fCFn, )

be a positive number. Then

be a

-unbounded sequence.

1&'71' ) N

for

nI

large enough. Using I) we have

SUp

C