Stability Problems for Stochastic Models: Proceedings of the 9th International Seminar held in Varna, Bulgaria, May 13-19, 1985 (Lecture Notes in Mathematics, 1233) 3540172041, 9783540172048

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

1233

Stability Problems for Stochastic Models Proceedings of the 9th International Seminar held in Varna, Bulgaria, May 13-19, 1985

Edited by V.V. Kalashnikov, B. Penkovand V. M. Zolotarev

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Editors Vladimir V. Kalashnikov Institute for System Studies Prospekt 60 let Oktjabrja 9 117312 Moscow, USSR Boyan Penkov Bulgarian Academy of Sciences Centre for Mathematics and Mechanics P.O. Box 373 1090 Sofia, Bulgaria Vladimir M. Zolotarev Steklov Mathematical Institute. Academy of Sciences of the USSR Vavilov st. 42, 117333 Moscow, USSR

Mathematics Subject Classification (1980): 60B 10, 60B99, 60E 10, 60E99, 60F05, 60K25, 60K99, 62E 10, 62F 10, 62F35, 62H 12, 62P99 ISBN 3-540-17204-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-17204-1 Springer-Verlag New York Berlin Heidelberg

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.

© Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210

CON TEN T S

Page

Zolotarev, V.M.

Preface

1. Hohlov, Yu. S. The estimation of the rate of convergence in the integral limit theorem in the Euclidean motion group 2. Kagan, A.M., Zinger, A.A., Contribution to the analytic theory of forms of independent random variables

11

3. Klebanov, L.B., Manija, G.M., Melamed, J.A. stable laws and estImatIon of their parameters • • . . . . . . . . . • . . .. 23 4. Koicheva, M. The method of metric distances in the problem of estimation of deviation from the exponential distribution .......•.......................•••.•.............. 32 5. Korolev, V.Yu. The accuracy of the normal approximation to the distribution of the sum of a random number of independent random variables .............•....•..•................ 36 Kruglov, V.M., Titov, A.N. Mixtures of probability distributions .....•.......................•••................. 41 7. Maejima, M. Some limit theorems for summability methods of LLd. random variables .........•.....••.•.•............... 57 8. Nagaev, A.V., Shcolnick, S.M. Properties of mode of spectral positive stable distributions ..••.................... 69 9. Nevzorov, V.B. Two characterizations using records

79

10. Nikulin, V.N. On orthogonal-series estimators for probability distributions .............••.•••.•..•............. 86 11. Obretov, A., Rachev, S. Estimates of the deviation between the exponential and new classes of bivariate distributions ....................••..•••••.•.•.•••........•... 93 12. Omey, E., Willekens, E. On the difference between distributions of sums and maxima ....•••.•••.•.•......•......•• 103 13. Popov, V.A. On the inequalities of Berry-Esseen and V.M. Zolotarev ..•.•..•••...............•......•.•..........•.. 114 14. Radu, V. Some fixed point theorems in probabilistic metric spaces ...•.•..............•............................ 125 15. Riedel, M. The asymptotic BIAS in a deviation of a location model · ......................•.•....•..•............•. 134

IV

16. Rossberg, H.J. Cramer's decomposition theorem within the continauation theory of distribution functions ..•••.•..... 145 17. Rychlik, T., Zielifiski, R. An asymptotically most BIAS-robust invariant estimator of location ..........•.•...... 156 18. Sasvari, Z., Wolf, W. Characterizing the distributions oft era nd 0 m ve ctor s X" I X2. X-3 by the dis t rib ut i 0 n 0 f the s t e t i s t i c (>;,- X , X., ) "

172

19. Siganov, I. On stability estimates of Cramer's theorem

178

20. Szeidl, L. On the estimation of momoents of regenerative cycles in a general closed central-server queueing network

182

21. Vostrikova, L. On

190

J

and their applications

22. Vsekhsvyatskii, S.Yu. On some properties of ideal metrics of order'fl ...........•..................................•.... 204 23. Yanushkevichius, R. On and sample variance ...

independence of sample mean

c ••••••••••••••••••••••••••••••••••••••••

207

PRE

f

ACE

The INTERNATIONAL SEMINAR ON MATHEMATICAL MODELLING AND STABILITY Of STOCHASTIC MODELS was held in Varna during the week of 13 to 19 May 1985. Such seminars on continuity and stability of stochastic models have been periodically organized in the USSR since 1974. The Ninth Seminar in Varna was the first one outside the Soviet Union and the second international one among these meetings. The first international Seminar ( sixth in the general order ) took place in Moscow in 1982. The Seminars are organized by the Steklov­Institute of the Soviet Academy of Sciences in collaboration with different other mathematical centres. The Varna Seminar was held on the initiative of the Steklov­Institute together with the Centre of Mathematics and Mechanics in Bulgaria and the Allunion Institute for System Analysis in Moscow. Looking through the contents of the proceedings of all the nine seminars one can realize that the Program Committee has never put strict limits on the spectrum of subjects treated. The explanation is that the organizers wanted not only a broad exchange of research information among the participants, but also to propagate the fundamental ideas and methods of a relatively new research area. In particular, the idea that every approximation problem can be considered as a stability problem in the framework of a relevant characterization model was a central one. Closely connected to this approach is the use of metrics in the space of random variables and their distributions, which leads naturally to a more close consideration of different questions concerning probability metrics *. As a result the participants in

the seminars formed always two groups.

The first group

* See also the foreword to the proceedings of the previous seminar, Lecture Notes, in Mathematics, vol. 982.

VI

consisted of active adherents of the above ideas, tending to materialize them in solving concrete problems. The second group included participants who Liked to learn more on these new approaches and methods. The traditional subject area of the seminars was extended in Varna by including problems on mathematical modelling in general. This was a quite natural decision stimulating the interests of experts of the two areas. The Organizing Committee ( Chairman ­ Blagovest Sendov, Vice­chairman V.M. Zolotarev ) received more than eighty papers from a dozen of countries. There were more then 120 participants. A substantial organizational effort was necessary to prepare the seminar and to publish these proceedings.

are due to our Bul-

garian colleagues and hosts for the nice atmosphere that they meeting. Particular

throughout the

are due to B. Sendov, B.Dimitrov, G.Tchobanov and A.

Obretenov. The editorial

was done

by

l.Boneva

V.M. Zolotarev

and

E.Pantcheva.

THE ESTIMATION IF THE RATE IF CONVERGEM:E IN THE INTEGRAl LIMIT THEOREM IN THE EUCLIDEAN MOTION GROUP Yu. S. Hohlov The aim of the present paper is to obtain estimate of the rate of convergence in the integral limit theorem in Euclidean motion group in the case of the stable limit distribution.

The estimate is obtained for uniform metric. To obtain this

estimate we use some ideal probability metrics in the space of random variables in Euclidean motion group. 1. The integral limit theorem in group

In this paper we consider the group transformation /Rei on

Y

If(

d

igin in

IR .

g(n):::!ft .. ffncan

1M (d)

U

SO

!It '

(d)

!J,

Yr + U,

+ ... +

,92, .. ,

u, ... Un_

,

(Y, U) , where

, . . . , gn '

then their product

,where lJ(n)==tJ, ...

lfn and

1

be a sequence of random motions. V.N. Tutubalin for

([1]) and L.G. Gorostiza ([2])

THEOREM 1. Let

g

is a rotation about the or-

be represented in the form g(n)::::( Y(n),U(n»)

Let now

3

which is the group of one-to-one

can be written in the form

If we have several motions

Yrn)::::

and

t

is the translation parameter and

d

nY(a') ,

!Rd preserving the orientation of the space and the inner

g

product. Any element

1A1(d).

Y2 ' . . .

for any

d

d= 2

proved the following result.

be a sequence of independent identically dis-

tributed random variables with values in IA1 (d) and satisfying the following conditions: 1) the distribution of

SO(d>

lIIl!asure on 2)

E

I Y.1 1

2 ",

Um)::::

u, ..'Un

converges weakly to normalized Hear

J

00

Then the distribution of

n

-1/2

Y( rn

converges weakly to spherically sym-

2

d fl 2 IR and n- Yen)

. normal d istribution on met r1C

an d

U ( n)

are asymptotically

independent.

In the paper [3J B. Roynette gave a new simple proof of this result. P. Baldi described the collection of all stable distributions on the group

J!tUd)

([4J).

It turns out that every such distribution is a composition of a spherically symd metric stable distribution on A? and an uniform distribution on . Now we can consider theorem 1 as the conditions under which the distribution of tne random belongs to the domain of normal attraction of normal distribution

motion

when we use the scale normalization of the translation parameter. An analogous res­ ult was proved by the author of this paper in the case of the stable

limit

dis­

tribution ([5J). Recently A.K. Grincevicius ([10J) slightly generalized the result of

the

paper

[5J

and

1.1 '

THEOREM 2. Let

got

rt , ::72

the

and

sufficient

d IR . The distribution of random motion g,=(Y"lI,J

belongs to the domain of normal attraction of the stable distribution on the paraeet.er

a, 0

«

2)

#Wrd) with

ex < 2, CY =F 1 under the scale normalization if and only if

1) the distribution of

so (d)

conditions.

be a sequence of independent identically dis­

. "

tributed randOlll motions of the space

necessary

Urn j converges weakly to normalized Haar measure on

,

P{IY,J>x}=(C+O(f))lx

cX

, as s:.... 00 for some C:> O.

Now it is natural to ask the question about the estimation of the rate of convergence to the limit distribution. In the solution of a number of problems of the estimation of the rate of convergence in the limit theorems and in the problems of stability of characterization of distributions the so­called ideal metrics were very useful ([6], [7]). We construct some metrics of this type on the group M4(dj in this paper and discuss its properties. After that we get the estimation of the rate of

to the stable distribution on group 2. Ideal metrics on the group

AWrd; for

uniform metric.

AWrd).

At first we remind of the definition of the ideal probability metric ([8]). Let E be a complete, separable metr i c space with metr ic

d

and a () ­algebra

J3 of

3

Borel sets generated by this metr i c , Let us suppose that we have an associative semigroup operation ,,0" on space J(

(E)

E. This operation induces the operation in set

of random variables with values in E. Let us consider also a collection

.iJ ={d} 'J e r}

of one-to-one and continuous transformations of space

E

and sup-

pose that the folluwing conditions are fulfilled: 1) set and

IIt 0 02! 2) 3)

r

is a normalized group, the norm of an element

I!, I .

I

J2 1

f

is denoted by

I el' ,

,

J2 ="v1, .lJdZ

£)1, ciJe

is the identical transformation of the space

4) for any

1(X

j

IE

r

X, Y e E

and

'1

oy)

E,

rX J 0

1 (Y )

DEFINITION 1. The simple probability metr i c in X ( E)

is called the ideal

metric of order S , S:= 0, if it has the following properties: 1) for any

X , Y,

which are independent of

Z

I'(XoZ, YOZ) /,(ZO%,ZO Y) 2) for

EIlY

I

to

r

and

£:

fl(%, Y)

X, y

I' (-7(%) '1 (Y») Itl Sometimes we shall say shortly that In our case we have

s

jI

·fI( x, Y) is the ideal metr ic on

E=Mrd),D={4,C>O}is

E .

the set of the scale normal-

izations of the translation parameter. The main result of this section is the following

THEOREM 3. let

if.,

be an ideal metric of order S on IR

following properties: (1)

j't(Y+UY"

(2)

;Ut( Y, + u, Y, >2 + £:

'2) Y) :: It, (1, ( Y) ,

SUPctf;+!lrx,

xclR

(Y»)

Xel!?

1

"

d

, which have the

4

g

where

(Y, U) ,

!J, = ( r; , If,) , .%'= (r; ,

are independent random motions. Then the metric

sup

==

, OM(dJ

X

where

are the random motions with distributions Ff and

and

probability metric of order

S on

IIrf

is the ideal

(d) .

PROOf.

be random motions and

independent of

!fa'

e

X'

.7,

and

are

e", !I e IA1 (d)

jI(!ls0 gt' !l3°!k =: BUp jI,(!I0§j0!l,(X') , go gJ o f2(XJ) x,g

Itt (!I

==

0

(

r; + u,:r) , go !Is ( >.? + 0 x»)

6sup;UtCY,+u,x,

x

!E: sup /'t (go!!, (X) .x:,fl

,

go g/i.l')

Here we use the property (1) of metric

=1' ({1"

JU, .

!/2)

Analogously, using the property (2), we

get

fI(!J,°f/a,

x.g

!E:SUPh(g°fl,(,x), gO!!;(,x))=j/(G;,g2)' x,g Further

fI (.11: (!it), 4

=x,y sup fit (j°.iJc ( flr )(X), go 4

= $UPjlI,(Y+U(cv,)+UU,x, Y+UCCiE)+ UU2 X ::e,g

= SliP

Y+UY,+ULJ, (; X»)J C(f Y+

oc,gF'f

C

Ie/

f/,(hog;(z),

s

= leI

(X)) )

U0?(; ,zo»))

$

We will use some metrics of this type in this paper. Let the variation distance, peetively. The metrics

Var,,p

and

?:"s

be

the uniform metric and the so-called Ss-metric on p¥dr es -

Var

ideal metric of order S on

and'p are ideal metrics of order S == 0 and It is easy to

Ss is

the

that these metrics have prop-

erties 1 and 2 of theorem 3. Consequently we can construct the ideal metrics on the

5

group /l,1(dJ using these metrics, which we will denote by the same symbols Var,

.

We remark here that for these metrics we can omit the supremum over

p ,

gEPWCg)in

their definition. 3. Some 1 _

In the proof of the main result of this paper we need some lemmas, which will be proved in this section. At first we introduce some notations. Let Q/r be the

a.

stable distribution on I/H (d) {

-i1

with structure parameter

is the projection of

LEII4A 1. Let

If, 1

tt

i\

t/

!R .

be the distributions on 1M (d). Then - il

*;::,

pRO(F. Let!l, =:(

and

on space

- /I

*

C(d,d).:1

and

flex

be random motions with distributions

-/I

QC{ •

.P ( P, , ) =.P (g,,, z g ) = SUp x P (g, UX') = ;l/P'p ( Y, + 6

cc and scale parameter iI ,

o, X, v., + U2 X

*' p',

-/I

Q(X

* Pz) +

(-

(X

A

/72

fit)· Var

0

(&:

g,2 (X) 6 A} d (ijOC\f if-

varCO: *

rj"

c- rAJ

+j>

tt)+ JJC

r/JZ)

1

0

+ Slip Slip II p{ gtI s:

J/.. =SUP d X e II? 0""-'"

/ m

t;+

1//;0 ... oy'd(fJ(xxdy)-fJ (x)Cdy)))

. "'d ,ldI? d +id=m

'i:x"

a

1

The main result of the present paper is the following r= [cxJ+! . Then

THEOREM 4. let #.=0,1=0 r-f Fl

.P ( P..

n

L

C(c;()

"

• d C( .a yrna x

[

] n 1- r/o: ,

(s, (p, Qo:) ,

PROOF. In proof of this theorem we follow in principle the proof of the analogous result in [7]. Let n

is fixed,

fj'

!j

and

uun

ions (!iWYU),UU»)and (j-l/CXnJ)j

are distributions of random mot-

Using lemma 1 we get

Further

Finally we get

jJ (

C '.p(

f}cx)

f;-

cn-j-rr/n

m

+C.;z::..fJ(P . ,Q J=f n-J-r ex

+C +0

) •

F

IX

(ij it;{. fJ ex

+c(d,c;(),1

-)I'

17-m-1

flex

I,

if-

I

IX"

(/trlfl)ln

o:

Yare ac(

- it

"" p, , Orr *

_ iI .i/rr - iI )·Var(Q *P*I} ,Q If IX 1 cc ex

.:fl!: O( iJ /! )(. a(f7-J-O!n;(- p..

J =0 J'

A

(17- n/n f'

(/117 oC'

ij A :/I' (X

+14 +4

or

)

aex "" aa: ) r/n-J-l )In*' din I}JII7) cx fin

Jln

If

(X

0(

!l il % [}(fl-m-l)ln;;

'cx

fin [}(X

a(trlftJlfl) ex

8

=[ ;

m

Let

J.

Using lemma 2 and the property of regularity of our met r Ics

we get

.,.

C

.I..

J

0 var (r/rl-j-I)ln p .i/n 1/ *" Q}=O, t , r e s J

J

In a neighbourhood I t I

.. K-I (K=1,2, ... ), i

the

Yp­strictly

Gaussian

Yp

THEOREM2. let the r vv,

Y

relation (2).Then the r v , i

(3)

is

r. v ,

has

the

CX,j3,A ,

Yp­strictly stable iffits c.f. is of the form

min (!

­L ­ f)

'ex:

tion (2). Then the r ,v ,

vp

})

,

vary in the range

(4) THEOREM 3. Let the r ,v ,

distribution.

have the generating function (1), satisfying

.I(i)=ho c-,1 I t l0 ' J':::12 J ,

and such that there are constants

infFe,y: I'min ( 'M)

with more than

points of growth for which

C, ,

C2 .

(1£)

There have been proved the following assertions in

which we state

here in the following form.

PROPOSITION. Let the d.f . F(x; U) the posit i ve numbers

t,

J ' .. ,

tk

have more than /( points of growth and

be such that

... ,k)

(1) the functions

are continuous w. r . t.

oe

7.i

and

'f

ZI!. ) ) j

=:

I, ... ) k,

have continuous w. r , t.

partial derivatives up to the second order inclusively; (Li )

Then

det 1l

iJz

(22 ;

X ) =f:. 0 .

7.i and the random vector

Vii (7£ - U)

is asymptotically normal

-1

N(O,lI (1fl;Xn If, moreover, uniformly w.r.t

rc»; 7.1) E:.1

and the estimator

u"

is

Vii -consistent

jr ,then the convergence of the d.f. of the random vector

27 the limiting normal d.f. is uniform w.r.t. the set

JC.

Let us apply the obtained results to study the properties of estimators of a parameter vector ZP.=(a,p,71J

of the

Yp-strictly stable law with c .f", (3)

n

Assume that there is a random sample of size F(X;ZR)

-I

of the law (3). Put

(IfnctJ)

Relf',,(Z;')*O

that

from a populat ion with d. L

>0

and let

be such

1,2,3,4

Using the random sample we can construct the MSM-estimator with components

based on the auxiliary estimator

o:"*=: In

(9)

if

Ilfn (l,) I I lJ1n ( l z> I 0::

if

The estimator

/ In 7:2

*

,

,in general ,

values from a larger set than (4).

ii:= (ct,ji,;J)

Therefore, let us introduce a new estimator

(10)

{"' A

a

= :

I

0< eX


l

k

fl

'k

g (iy») +

m

k==f

k

as the product L(2:>= QCZ)·

L(Z)

-6y

+

[J

Jk

)

e

i(Z)

. where

Note that m-f

I

(13)

holds for

lex t {]

zr -

y:?: 0 . 00

to infinity as

•

m-f

m-1

(iy) + !fm m

It was assumed that

g

m-1

.

(ty)

lex t F::: -

00 •

Thus by (8) we have that

According to lemma 4 the functions .f ( /r: and co

g ( iy)

tend

• The relations (10) and (11) imply the boundness of the

46 ratio griy) / .fUy)

. Tak i nq (12) and (13) into consideration we get

sup ariy) = sup (/L r iyJ/ / T(iy))

y,=O

I ,e, function

linear

a is

!. Equation (17) takes the form

11 (z)=/exp{iazu-

(20)

UJ 2

z

2

}dA(u)+I(Z).

0-0

In (20) the number Ll

is positive. Really, if Ll ==

0

then the left-hand and the

right-hand sides of (20) have different orders of growth as Iy)-+oo. If, for exam-

ple,a>O then put Z

= iy

and divide both parts of (20) by II (iy) . Let y

tend to

infinity. Taking into account (18) we get the contradiction: /=0. 2 2 Fix a number C such that C> ();/ J ,A (C) « ! 2

C/ y2}

. The inequality +l(iy)

obtained as a corollary of (20) is contradictory for

y

large enough because the

order of growth of its left-hand side is less than that of the right-hand side if

y-;>

00 •

2 2 for U>u /Ll

Consequently, A(u) ==!

have

!

lim _1_._(

y ...... -oo /loy) .

lim

y""-

00

1

. Besides that by (18) and (20) we

u71+/J

I

hV(iY)dAcu)+l(iYJ)

0-0 2

2

((2

2

- . - exp{lexly(];/J + 2 Y }==O •

/lUy)

This contradiction is a result of the assumpt ion that function points of growth.

N(v)

has not any

51

Now consider the second case, i.e. the case when the function

lV(v)

has points

of growth. We have already marked the fact that the d. f. A cannot be degenerated at zero. So there exist positive numbers C and c"C 1

{}

is con-

is constant in the interval (0,1)

V=!

at the point

. However the supposition

h(Z)

tradiction as it was shown above. The ch. f.

,otherwise /\I(V) /\I(V)

== 0

• It

== 0

,

leads to a con-

from (17) can be represented

as 2

htz:

(23)

.

Fix a constant 0 such that A(O):!Iq .

According to (17) and (18) we get

00

I;:: Zim -'-.-!hu(iy)dAO/) y...-co 11 (ry) C

/II q

we get respectively

00

and

.

L]2::lIt:{

and the representation of

n/l1fJ

0''l8 .".

the proof of theorem 3 in the case

A=E;l.lfj

LI is a positive function

As

1 for all Y large enough. For such y by (17) we have

is greater than

+ /Z(iyJ! Letting

.

,

cr.> 0

and

13

,where ;j >

. Fix e ,O< 0

can be taken arbitrarily small, we get Now prove theorem 3 in the case

v.> 0

j3 •

integrable over the normal d. f. for any integrable over the d. f. the functions

N(v)

A

£)./({'

and A = 0 • The function It follows from (9) that

for some positive U and all

flU

. As

j3 .

SOft 0

2 2 for U> () / Ll and A > 0 we obtain

and a= 0 :

a )1/2exp] - a!J-n;2 } o·(n)= (- J 2rrn 1 2n

, Dej
} be a random-walk method. J

(0

-1/2

s!lpCj(n)/Vconst.xn

.

J

r>O

(E) Let

and assume

P

Elf,! for some p>1Ir-!

when r0 ,

0 ,

(a)

E

"

(I +

log

+

-2

2

114

J -c

OQ

-1/2

lim sup

(c)

lim Slip (411n) (d ZDgn) n-oo

C41ln) cd logn) 1/4

={ p(1- p)} This was

-fl4

-1/2

n

1

also extended to

j

J-O

a..s , for some (any)

=1

x.) J

J.

J:;O

2

J

-n

0

(b)

p

E

av s ,

(f-p)

n- J

(0,1) .

random-walk method by Bingham-Maej ima [2].

the

THEOREM 6. ([2]). The following are equivalent. (a)

= d 2> 0

EXa = 0 ,

£exu"U+loq+IXolf2J (b)

lim sup (41ln) n_ oo

1/4

, 1 .

62

where Zo:- is a stable random variable with index ex and L (.) is a slowly varying function.

THEOREM 7. ([13]). Assume (A) and let 1

--

(4 2)

•

00

...2:' C·

So,)j:=o

(A) X

J

J

I?

with some normalization a (/I) > 0 .Define

i!i I 8(i!) I 00

(4.3) when

0 < 0

.1 O:[.(}t)

-tjlA

XJJf>O

J=O

1;00e -tudZ(VJ,t>O }

}

a

0

in C [e,=) for any s >0 . To discuss the functional limit theorem for the random-walv method, we start with the following.

t

THEOREM 13 ([ 9] ). Let

and

be two sequences of random variables ,2

2

generating two random-walk methods p(f', () ) and p( u;U ), and set ,

n

I

J

S=.;£'f. . n J=1 J (I ) If

!' =/-" ,

(Li )

n

If

,

then

1/2-1/(2«)

(6.1 )

17

,sn =J=1 ,;;;:£

Lt n

1/2

I' =PI"

)

1/2-1/(2")

00

01=0

P(Sn=j)-Xj

J

n

Lr n

1/2

)

0:>

J=O

P(Sn'=j)AJ)

,then the limits in (6.1) are independent.

COROLLARY 13.1. n

1/2-1/(2a:)

and

have independent limits. Hence the Borel and Euler sums generated by the same sequence have independent limits. The following statement is essentially the same as Theorem 13 (ii), and it

66 gives us some answer to the question of functional limit type for the Borel and Euler sums. THEOREM 14 ([9]). If t

n

'* S,

then

1/2-1/(2a)

[nt]

J=:O

=J)X J

and

n

1I2-1/(2c()

LCn

'/

2)

QO

J::o

J)X J

ens]

have independent limits. COROLLARY 14.1.

Define

Ii 1/2-f/(2 0 .

J

t

=1= s

COROLLARY 14.2. Define n '12 - / ( 2c< )

Lln(tJ=

' Lcn'/2)

tnt)

0

tnt:

J

( j )P (1-P)

CntJ-J

XJ , ,t>O.

Then Ll (t J and L1 ( S) have independent 1 imi ts i f t =t= s n n

.

7. THE RATE Of CONVERGENCf IN THE STABLE LIMIT THEOREM. In this last section, we shall give the rate of convergence in Corollary 7.1. for a restricted class of random variables

{X } J

in terms of an ideal metric,which was recently defined

by Maejima-Rachev [11]. We start with the definition of this new ideal metric. Denote

Ifgll p = 1/ glloo Let

r >0

00

p

Ig(XJ! dx]

tip

)

P
I ,

P( y

.r) aa:

81' ( X, y) & { k, ( X, y) } Y can et. xk,,(X, Y)

where

00

k (%, y) = r r /

, X I1'-1 I P (%

r

X) -

1 ,

I

-00

We restric random variables as fallows. Assumption (B). Suppose 0 < ex < 2 ,and that

of attraction of a stable random variable 1:

ex:

= exp {-I W lex} . THEOREM 16. Let (i) O.c: ex =-It J exp(-i

signt) ,

OltlE, jlfJ(f)j=exp(--Lldx'sin rr;e):; exp(_ 4 Jt j1/2) ex E using (17) we obtain (33)

In addition we have (34)

Combining (29),(30) and (32)-(34) we obtain the statement of Lemma.

75

Note that the integral in (28) is real-valued for any integer part icular

It 2Zn t ex p C-

k

. In

0

2:

00

f!r(X)=-;

1t 00

1

- 211

t)sin(tx+fZnt)dt

(J

2(

Jl2 2 if-Zn t)exp(

TC zt)cos(tx+tlnt)dt

o Proceed to the proof of Theorem 2.Denote by

Z and rna the modes of

ex

(X) and

g(X) respectively. Firstly we show that

z = rna +

(35)

W (£)

,

Choose .z;..cx2 X2 '

••• the

are independent? It is easy to prove the following

lemma. ... ,Xn

lEtIfA 2. Let

8 ;

...

fz" '" f n :=

i

-00

be independent r vv, with distribution functions

FIXi, for2,

F and S< being continuous d.f . and (Xi>O(i=1,2, ... ).Then the r ,v ,

k

are independent,Pff/ 0

in theorem 3 can have any positive value.

3. PROOF OF THEOREM 1. Consider the equalities

One can rewrite (1) in such a way:

x

00

j

(2)

-00

H2 ( X )

J H,(YJ!k (yJ dy d Fn(XJ

-00

00

=! where

_00

co -00

H,(YJ0«Y)H2(YJdFn ( y ) ... F,,_1(X)( ifk=:n-!

f?(:x)=F,(X) ..

,

,then

). The can-

ditions of theorem 1 allow us to substitute in (2) any absolutely continuous d.f. ChoosingF"(X)=F,,,e(XJ=(X-Z+f)/E Z (3)

(-= , 00)

H

2

for Z-e£X!:Zand lettinge-+Owe obtain for

Z ( z )/ _

Z*).

(4) _00

The next relation follows from (4):

II; t z:

C

(Z)

(5)

- - - -- --- 0«Z) - 7-C 111 lZ) -

flk ::: CI(t-c)= i

where

k

H,(Z)

(z >z*) ,

f4(YJ:{rYJdy /

il7z) =!k II,fJk(Z)

One gets now that

#,'(z)

j3,

-

. 00

for Z > Z *,

(6)

(Z)=(F,(Z)F;(Z)· .. Fk_1(Z))

which is valid for Z >Z ... From (6) F;(Z) ':

(7)

withZ")

0

,

84 R£FERENC£S

1. Renyi A. On outstanding values of a sequence of observations. - In: Selected Papers, Budapest, 1976, p.50-65. 2. Nevzorov V.B. On record times and inter-record times for sequence of nonidentically distributed random variables.- Notes of Sci. Semin. of Leningrad Branch of the /otlth. Inst. 142. (1985), 109-118. ( in Russian) 3. Tata M.N.

On outstanding values in a sequence of random variables. Zeitschr.

fur Wahrsch. verw. Geb. 12, (1969), 9-20. 4. Ahsanullah M. Record values and the exponential distribution. - Ann. Inst.Stat. Math. 30 (1978), 429-433. 5. Ahsanullah M.

Characterization of the exponential distribution by record values

B41 (1979), 116-121. 6. Ahsanullah M. On a characterization of the exponential distribution by weav homoscedasticity of record values,.

(1981) 715-717.

7. Ahsanullah M. Record values of exponentially distributed random variables.-Statist. Hefte,22 (1981) 121-127 8. Ahsannulah M.,Holland B. Record values and the geometric distribution.-Statist. Hefte,25 (1984) , 319-327. 9. Dallas A.C.Record values and the exponential distribution.-J.Appl.Probab. 18 (1981), 949-951. 10.Gupta R.C. Relationships between order statistics and record values and some characterization results.- J.Appl. Prob. 11.Kirmani S.N.U.A., Beg M.l.

l! (1984), 425-430.

On characterization of distribution by expected

records.- Sanvhya A46 (1984), 463-465. 12.Korwar R.M. On characterizing distributions for which the second record value has

a

linear

regression

on

the

first.-

B46,

(1984),

108-109.

13.Mohan N.R., Nayav 5.5. A characterization based on the equidistribution of the first two spacings of record values.- Zeitsch. fur Wahrsch. verw.Geb. 60 (1982) 219-221. 14.Nagaraja H.N.

On a characterization based on record values.- Austral. J.Statist

(1977), 70-73.

85 15.Nagaraja H.N.

Record values and extreme value distributions.-J. Appl. Prob. 19

(1982), 233-239 16.Nayav S.S.

Characterizations based on record values. J. Indian Stat.

1981, 123-127. 17.Pfeifer D. Characterizations of exponential distributions by indepedent nonstationary record increments.- J. APpl. Probab.19 (1982). 127-135 (Correction:

.!2.L 906 ). 18.Srivastava R.C. Some characterizations of the exponential distribution based on record values.- Bull. Inst.Math.Stat

l

(1978), 283

19.5rivastava R.C. Two characterizations of the geometric distribution by record values.- Sanvhya, B40, 276-278. 20.Srivastava R.C. Some characterizations of the exponential distribution based on record values. - In : Statistical distributions in scientific work, v.4, Dordrecht, 1981, p.411-416. 21.Srivastava R.C. On some characterizations of the geometric distribution.-Ibid, p.349-355. 22.Taillie C.

A note on Srivastava's characterization of the exponential distribu-

tion based on record values.- Ibid, p.417-418. 23.Westcott M. Characterizing the exponential distribution ( Letter to editor ).(1981), 568. 24.Khinchin A.Ya , The works on mathematical theory of queues. Moscow, 1963 ( in Russian ). Department of Mathematics and Mechanics Leningrad State University Leningrad

Received 20 May 1985

ON ORTHOGONAL-SERIES ESTIMATORS FOR PROBABILITY DISTRIBUTIONS

V.N. Nikulin

Let X =

{'3" } , j = I,. . • , n

be

independent identically distr ibuted random

variables with distribution function

F to be estimated. We consider the ortogo-

nal­series estimators of

F given by

a: p{jefJX)dt J

_

00

where

1

/

P ct,X)= - {j 2 n» -(j

Un''} l/l(X.-t) J

";=1

d;t

,0-6U

2 BXp (- X - Y )

LIn

-

Y

x

/ 6'n(A;

Y

or Lln(X,y)---:> 0

(X,y) E

N13U

fJ" (U, v) dudv

and be with the same exponential margin-

x y als with parameters Af ( U) , i/Z ( n)

such that

-1

-f

aUO(n) = i1f

( n J A2 ( n )

• If

has finite Laplace transform, then -1

lim Cn (/1 f if and only i f

PROOf.

(fl)X,.4

lim

-2

agO

(n)

The statement

-1

2

(n;y)

= exp (-x-y)

a ,1( n } ==! • follows

from

Lemma 2

and the proved above Theorem.

Further, we shall estimate the deviation between ad. f. bivariate

on U.

"0 00

6n (X,y) = auo / /

F" (oX, y)

/

and from (9) we conclude that (8) is true.

COROlLARY 1. Suppose -f

X

1

CD 00

. Therefore

exponential

d.

f.

with

F to fiR

and a

[see

parameters

(n].

In order to get the required estimate we shall prove some preliminary lemmas concerning the inequalities between metr ics in the space

:;: of all d.

f.

I

S

on

2

!R+ = [0, 00 ) x [0, 00) . LDI4A 3. for any d. f.

f

S

;:;

and

":?

on

IR:

such that

marginal densities

M= SUp

If

ax

f2(x,OO)I+ Sup ldd Fz(oo,X)1

x

the following relation holds

where.fJ is the uniform (Koloogorov) metric

I

SUp !j'(ZJ-f2(Z)j , Z= (.r,y) Z IE 1R 2

f2

has bounded

97 and L is the Levy metric

The proof is evident.

jf

Let

be

the

following

metric

on

the

.f

space

of

d.

f. 's

2 on $+

/,O:;,"2)=SlIp{lj.fd(F;-Fz)):I

7f} ,

2

where

IR+

de is the class of functions

which have continuous derivatives

aU)

x

=

max

U,JJ ,J('

LU;, C=

ax 8y2

F;) 8

CIf 1/5

4- ( T)

PROIF. Let

that for some

a

such that the functional

(/+j .I

. .

8x l 8y J

F,

1/5

,

(F,J2

and

F2

on

IR;

the following estilllate holds

) ,

=4,5526081

L ( F, J;) and

with real values

1 .

LEfIoIA 4. For any d. f. 's

where

a"f 2

2

IR+

= { (/,0) , (0, 1) , (1,1) , (1, 2) , ( 2, 1) , ( 2, 2)}

I 0 (/) I 6

satisfies

I

max

J(:J:,y),(X,y)

:>

e

. Hence, without loss of generality, we assume

6

Further we use the main idea of estimating

fI

in terms of L , given by G.

Yamukov (1977) [2] (see also V.M. Zolotorev (1976) [3]). Denote

.B =

J x [0,6 J , .B (c) = [0, a + eJ x [ 0, 6 + e J

and

Jex,y)=

T12

[(-(oX

2

2

2

+Y )] .

It is easy to verify that

\0(1)1 6

-:

and

J !ex,y)dxdy

!

.

98

Consider the functions: lI(X,Y)::

(J /1I{(lJ"V2) e s ;

a -o; rr:

)}j'(2

J1?2

z ,2 -E-)du,dv e Y-V

+

and

p(X,yJ

2V(X,y)-I.

vrx,y)

Obviously 0

, so that -I

1

, i f (X,y) belongs to

fjJ(X,y)={ 1 -{

0

I{(V"Vz ) £ .B(

we get

f

and from V,

Suppose X

ff )} = f

i2':

a

0

I { .}

1 ,

.Il

if(x,y) does not belong to

Really, let (X,y) E B(e) j =

P(X,y)

. Therefore II ( x,y)

+

B( c) .

e

. When

=f

then from U, "" f (X,y)

e B

follows we have

which gives If (X,y)

:= f

Further we find

8rp

ax so that

Ik8x l

2

LE

/1

1 1

o

f

0

-E.L I du1 du.2 aX

Analogously we get 7

11: e

ay

I

a3

I

Therefore

fI(lf)

(15)

h (X,y) o

;e3

3 2" 2 '2

= ;

eft.

=

,

4 a

:I(.!) = 111(X,y)d C8f ( x , y ) - 82(x,y)] . o 0

Integrating by parts we get 00

(j(f)=jJc.:r,O)

o

Next

:;, =-/ o

00=

ce, (X,O)-62 (x,O))dx-j / 0

r:

0::>

j( x,D) d

[8, (X, 0)- {]2(X,in] =

J [6',

= /(0,0) [6,( 0,0)-62 (0,0)J+ and

0

o

)-{]2 (x,O)J

da:

dxdy .

100

CO

.1 __/ [ / OO dlrX,y) dx [C,-C,J]dy 8y

2-

()

CQ[ 8I(O,y)

=;

(}

o

y

8xa

/

y

dxdy ==

00

0

elf

j j( 8,-(J2)dvdv 00 00

8xd d Yxy

,

2

8xdy the

and

R(X,y)

same manner as

=:

in

(;,(X,y)- IJ2 ( X , y ) the

0000

0 00

+/R(O,y)

o

00

0

a

00 00

'9';O,y) dy+/ jRex,YJ y

0

R(x,O)

a2 (

0

y

a

g;x,O)

a:

dX'dy

Hence 00 00

J(fJ = ; o

j.f( «r: d [ IJ, (X,yJ - [;2 (X,yJ ]

0

00

+/ o

(6, (O,y)-6'2(O,y)) 2

+

a8x8y 'f(O,O)

II

0000

8y

dy

0000

o 00

81(0,y)

we get

representation of

/ j !I(I,y) d R( x,y) =:!l( 0,0) R (0,0) + j

o

d d .x y

0

a 'f(x,y)

Using

j/

00 00

=/ !!1(X,y)dR(X,y) o 00 ""

g=

2.f ;00/00(c,,-6'Z) ox8y a

0

a2.f

00

where

2

(B,(O,y)-G2(O,y))dy+

ca co

finally

,

0

(b',(X,y)-IJ2 (x ,y )) d x dy

0

3

+ j[jj(6.,(X,VJ-6.2(X,VJ)- a fO}

11 FxGII 611FII UOIl • For m

m

m

the function m we shall often assume that m satisfies one or several of the following conditions:

m

J:: lim x-?

00

sup Iy!";; h

m e OX: Zimsup x., 00

m(x+y)

m(x)

sup ly/6.h

m(x+y)

m tx )

for

!

O

all

h >0

106 :II:

/ m(x-y)m(y)dy o =2jm(y)dy m t.x ; 0 OQ

m

G:

SlJ: lim

m x m(x) = lim mt x )

.

:x _

The

x m(XJ

m

m e OS]) : 11m sup


0 between two bounded on (-""',co)

F and 8 is defined as the Hausdorff distance between their complete

functions

graphs F and fJ :

r(F,B;a)

(4)

When The

Q'

r(F,C;a)

"'! one writes

Hausdorff

simply r; F, (])

distance

between

sets

was

introduced

( see [6] ). The Hausdorff distance between functions

by

F.Hausdorff

(see the definition (4))

was introduced by Bl.Sendov and B.Penvov [7]. In the theory of approximation the Hausdorff In

distance

was

systematically

used

by

the case of distributions functions

Bl.Sendov

(

see

[4],

[5]).

the Hausdorff distance ( with

parameter 1 ) is vnown as the Levy distance and is usually denoted byL(F,B)([B]) The definition of Levy goes as follows : let F and G (5)

L (F,

B)

= sup X

in! {Y:F(X-y)-y

B(x) £:

be distributions. Then

Fe x+ y)

+

y}

(- co,ee)

It is not difficult to see that (6)

UF,fJ)=r(F,G). The corresponding definition of the Levy distance of parameter 0: > 0 ( the

Hausdorff distance ) is the following: let (7)

r(F,{]j(X)

= L(F,[]jGr) =

F and G be distributions. Then

sup XE(-oo,oo)

inf{y:

116

- y + F (x - ex y )

Fe X + «y) + y }

(; (X)

We shall denote as usual the characteristic function of the distribution F by

.I

!(t>=jeitxdFe X

)

- 00

and

the

characteristic

function

of

the

distribution

G -

Our aim is to obtain an estimate for ru; Gja)( L ( F, G; Let us recall the well-Imown Berry-Esseen inequality

TI£OREM A. Let the distribution

F

0:»

by

g

by f and

g

[1],[2] for ,p(F, G)

satisfy a Lipschitz condition with

constant A . Then for every T>O we have :

+

,jVCF,G)

(8)

[1],[2].

The

Lipschitz

I F(x+h)-F(X) I

r

i- / o

condition

1I(f);9(tJI dt with

A

constant

means

that

A I hi.

Zolotarev's inequality for the Levy distance L (F, B) is the following [3] :

TI£OREM B. Let F and (; be distributions. Then for every T> 1,3

we have:

r (9)

LCF,B)

l;i +

/

!ICt)/(t)/ dt

o Notice now that there exists a connection between the Hausdorff distance with parameterO'>O and the uniform distance. Let us denote by w(F,'

g), cJ > 0,

the

modulus of continuity of the function F : w(F;8)= sup

{I Fcx+h)-FCX)!

:

E

(-oo,oo)} .

In probability theory the modulus of continuity is usually vnown as the function of concentration of the distribution F( see for example the book [9]). It is easy to see that the following lemma holds ([5],[10] ) : LOttA1. For everya>Oand for every two distributions

r( F, B ; cr ) 6.p ( F, G) 6 r CF, G; ex) +

w ( F, ex: r( F, G; ex») •

If the distribution F is continuous, then

(10)

lim r(F,G;o:) =,pCF,G).

F and B we have:

when

and therefore

117

The inequality (10) shows that the Hausdor ff distance with parameter ex' > 0 can be considered as a generalization of the uniform distance. But on the other hand it is clear, that it is not possible to obtain the inequality (8) directly from (9) and conversly. As we have already mentioned our aim will be to obtain an estimation for r(F,Bja) , from which

both estimations (8) and (9) follow.

Also from our estimation will follow the result of A.S. Fainleib [11], also [12, p, 94], which can be considered as a generalization of the Berry-Esseen inequality (8):

THEOREM C. let w(F;o) be the function of concentration of the distribution F. Then for every

r>o we have:

r +J IIU)-tq(t)/ --dt} C{W(F;y) 1

(11)

o where C is an absolute constant. If F satisfies a lipschitz condition with constant A ,then W(F;.!...)f:..1..r 7'

therefore (8) follows from (11) up to the constant. In the

of Fainleib [11), as well as in [12,p.94) the inequality (11)

is written in another, but equivalent form. The function h

.$(h) F

::=8UP X

c (- oo, oo )

-'-jIF(X+-U)-F(x-u)l d u 2h

o

is introduced and then the inequality becomes:

I f(tJ-g0 the following

inequality holds:

In (e +ex 1 W ( t)) (13)

+/

r

where C is an absolute constant.

1

!hiJ;g(fJ}

a

dt} ,

Before proving theorem 1 let us mal/-t

k e (O,JJ is fixed.

where

The main result in [14J reads as follows. THEOREM 3A. Every C-contraction on a complete Menger space (,5',

if , Ali TT

)

has

a unique fixed point, which is the limit of the successive approximations.

The proof of the above result is obtained from the deterministic Banach prin-

$

ciple, by constructing a metric on is such that f

which generates the

ie

,A

)-uni formi ty and

is a contraction with respect to that metr i c ,

As a matter of fact, the same proof is valid for a larger class of t-norms. This is due to the fact that the two-place funct ion d constructed in [14J is a metric in any Menger space ($,.7,7') if THEOREM 38. let ($ ,

d t p, C!) =

(6)

Slip

..r ,

{t,

T,.,,)

1m '

be a Menger space and define

FPq(t)

1- t}

.

Then (i)

d

is a metric on ;), which generates

the (c , A)-uniformity; (Lr ) Sis j -complete iff $

is

d -complete;

(iii)j:$-+$ is a C-contraction iff J is

d contraction.

PROOF. (i) We will prove only the triangle inequality. If

dcp, 1/) -c t,

rpqU,);::;.

and d(q,rJ -c t

!-t"

» 1- t 2

(t, + '2) which shows that

dt p.r)

2

":n( If,q 1- A

p=q .

LEtfoIA 3.3. Every C -contraction is (uniformly) continuous. PROOf. Let E> 0 and A

ko < min

ce , A)

e (0, f)

be given and choose

• Now if (P,q) e N( 0,0)

6>

, then

0 such that

'Pr';c cS) :> !- 0

.

Since

132

f

is a

F;

.i

p.;q

C -contraction, then we have that

F,

(6')

;rp.fq

(/(0) >/-ko >/-A

Therefore

F J. (krJ) > 1-/(0 .Ip q

, that is (fp,.fq) E N(E,A)

and the lemma follows. Now we can prove the following THEOREM 3C. S l/P

a0

n

J

0

and A t ( 0,1)

be given. Then by Lemma 3.1. there exists

such that (7) holds. Now i f

!P..

P =Pm

P. (E)::: F n n (c) > 1- A .f Pm! Po

n-ern n

Therefore that

C/o

P n

{p,.,}

J

and

/I rr

n (cJ

then we have

A) J V rrr

is a Cauchy sequence. Since $ is complete then there exists

converges to

qO'

is a fixed point for

From (9) and Lemma 3.3. it follows that .ff/ =- lf

f .

By Lemma 3.2

qo

o o

1

106$ such ' that is

is unique and the Theorem is proved.

REMARK 6. It is clear that the above proof holds in any complete probabilistic metric space, i f the (E, il) -topology exists, a situation totally different from what happens for probalistic contractions. REMARK 7. From Remark 5 it is easy to see that the classical 8anach principle is a consequence of Theorem 3C.

REFERENCES [1]

A. T. 8harucha-Reid, Fixed point theorems in probabilistic analysis,

Bull.

Amer-Math. Soc. 82 (1976) 641-657 [2]

G.L. Cain, Jr. and R.H. Karriel, Fixed and periodic points of local contractions on PM-spaces. Math. Systems Theory, vol. 9, No. 4 (1975-76) 289-297

[3]

Gh. Constantin, I. Istratescu, Elemente de analiza Ed. Acad. RSR, 1981

§i aplicatii.

133 [4]

O. HadHc, On the

(E.,

A) -topology of probabilistic locally convex spaces.

Glasnik Mat. 13 (1978) 61-:'6 [5]

PM -spaces ,

O. HadHc, A generalization of the contraction principle in Review of Research [lb. Radova] Prir. Mat. Fav. 10 (1980) 13-21

[6]

P. Mostert, A. Shields, On the structure of semigroups on a compact manifold with boundary. Annals of Math. 65 (1957) 117-143

[7]

D.H. Mushtari, A.N. Serstnev, On methods of introducing a topology in random metric

[8]

spaces.

Vysh.

Uch.

Zav.

Math.

6(55)

(1966),

V. Radu, On the t -norms of HadHc - type and fixed points in Review

[9]

Izv.

of

Research

[lb.

Radova].

Prir.

Mat.

Fav.

13

99-106

PM-spaces.

(1983)

81-85

V. Radu, On the contraction principle in Menger spaces. Analete Univ. Timivol.XXII (1984), Fasc.1-2, 83-88

[10]

B. Schweizer, A. Skl ar , E. Thorp, The metrization of

8M -spaces. Pacific

I. Math. 10 (1960) 673-75 [11]

V.M. Sehgal, A.I. Bharucha-Reid, Fixed points of contraction mappings on PM-spaces, Math. Systems Theory 6 (1972) 97-100

[12]

A.N. Serstnev, On the probabilistic generalization of metric spaces, Kazan Gos. Univ. Uch. lap. 124 (1964) 3-11

[13]

H. Sherwood, Complete probabilistic metric spaces. L, Wahr. verw. Geb. 20 (1971) 117.,.128

[14]

T.L. Hicks, On the theory of fixed points in probabilistic metric spaces. Review of Research [lb. Radova] Prir. Mat. Fav. 13 (1983) 69-80.

University of

Received 10.05.1985

Blvd. V. PSrvan No.4 1900

Romania

THE ASYMPTOTIC BIAS IN A DEVIATION Of A LOCATION MODEL M. Riedel 1.INTRlDUCTION.Consider a location model which is given by the parametric family

veRt

J={

where

F

is a fixed distribution function (d.f.) on the set

of real numbers and

tE IR.

(1.1)

In robust estimation theory the parametric model is not supposed to be exactly true. The observations

x,., X2 "

used to estimate

•• ,

the unknown parameter

V

are

assumed to be independent with a common d.f. which lies in some appropriate neighbourhood of the d.f. F

v

.The d.f.'s of this neighbourhood are interpreted as the

d.f.'sof the observations which are somewhat contaminated. The degree of contamination is measured by a number

e , O:!':

C

said to be an £ -neighbourhood of d. f. ;:;;

f{, .

ed a deviation of d. f. define a deviation

If = U

} of

HE (

1

. Then a class 11£ . The family

If the deviations of

.:F

11

11

0:6 e

of d. f. !E

I

S

is

I} is call-

for each velR are given"we may

by sett ing

r;,.)

tJ' IR Examples of deviations are the so-called gross error model and deviations based on neighbourhoods generated by some probability metric.

K

Let

be defined by (1.1) for K::: F

t

K(t) : == f- K (-t +) ,

E

IR

,

(A-2)

lie

(A-3)

H(K)=nH(K),

cf

E

(K) c He ( K) , 2

a: >£

7

IX

(A-4) (A-5)

8

'

1 (/() = 1(K)

,

by

For a general deviation we need the following

assumptions: (A-1)

and introduce the d. f. K

£2

,

135

(A-6) For each d. f. (J,

{t :

E

HE (/() }

is compact.

The assumptions (A-1), (A-Z) and (A-3) should be satisfied if preted as an

-neighbourhood. The assumption (A-4) means that the

is inter-

c

-neighbour-

c-

hood is equivariant with respect to any translation. By assumption (A-5) the

K

neighbourhood of the d.f.

c

can be easily obtained from the

-neighbourhood of

the d. f. K • As estimators of the locat ion parameter we choose statist ical functionals introduced by von Mises (1947), which are currently used in the theory of robust estimation. Denote the empirical d. f. of £n: = (%, •

r

the set of all d. f. • s , The functional

by

defined on a subset lJ

r

tional

r,

r

II,

contains

7!

where

{Tn: n = "

i(K)(

with values

n

is meas-

r (E £n)

T.' n ' --

2 , • . . } of est imators of the

• I f the observations X" %.2""

are

is wealdy continuous then { in}

independent with common underlying d. f. K and i f i is consistent to

jI be

and all empirical d. f. "s , Given a statistical func-

we may introduce a sequence

locat ion parameter

Let

n

of Jf

on JR will be called a statistical functional if the urable and i f lJ

C£.

see Hampel (1971) or Huber (1981), Proposition Z.6.1).Then

it is convenient to discuss the quantitative large sample robustness of the behaviour of its asymptotic values

irK)

terms

.For this reason, it may be use-

ful to consider the maximum asymptotic bias

r

of the stat ist ical functional criterion

in an

e

-neighbourhood H£ as a robustness

.

Since statistical functionals are used to construct estimates of the location parameter they should satisfy some add it ional conditions. Here we require that the statistical functionals are translation equivariant and antisymmetric. For further conditions on statistical functionals we refer to Bi ckel and Lehmann (1975) and Staudte (1980). Note that a funct ional T is translation equi variant i f for each

J'

E:

that

IR we have

i ( K) = - i ( K)

irK)

+J'

.The antisymmetric property of r

. Here and in the sequel

with respect to any translat ion and symmetric, i ,e . lJ

r

means

17; is assumed to be closed

:= {

Jt :K e 4 .j't' R }={K: Ke 4}.

136 Let jP be the set of all translation equivariant and antisymmetric functionals containing

in their domain of definition. Our main purpose is to determine

the minimum risks

4( e, hi):

Bt(c,hi): = inJ{

in! { 6 ( e, 7') :

6(8, 7'): 7'(£ j ,

reF) = o}

The second risk des-

cribes the case where the true location parameter of the d.f. ed by a functional

F ,which is measur-

7' , is zero. We remark that this risk is well defined if there

does not exist any symmetric d. f. $ we

and

7' .:f }

such that F =

for some

I

=P 0 . Besides,

will derive the most robust functionals which attain one of the minimum risks. The location parameter

family

J

zr

can be extended to the

e -neighbourhood

of the

in the following way. Define the minimum location parameter of the d. f.

K in the

E -neighbourhood

hi by

Similarly, the maximum location parameter of K in HE is given by

at i«, e , 1):

sup { ! : )( c

1 ( fj.) }

These location parameters considered as funct ionals of K are translation equi variant but not necessarily antisymmetric. However, if the d.f. F is symmetric, then the functionals

are antisymmetric. As we shall see the minimum

can

be expressed by means of these extensions of the location parameter. In the

case where the d.f. F

is symmetric and unimodal, Huber (1981) has

shown that the median is most robust with respect to B, ( e ,

1-)

{1t

where

is the

gross error model or the deviation induced by the levy metric. Our aim is to get rid of assumptions concerning the unimodality and symmetry of d. f. F • In section 2 a presentation of the minimum risks by means of the minimum and maximum location parameter is given for general deviations

of!f and arbitrary d.f.'s

F .

Section 3 deals with the minimum r i sks with respect to deviations based on metrics. An upper bound which is given in Theorem 3.1 turns out to be strict i f the d. f.

F

is symmetr rc , Moreover, it is shown t hat Huber's result cannot be

extended to location models with symmetric d. f.

F . For deviations

based on the

137

Kolmogorov metric and the Levy metric, respectively, the minimum

can be ex-

F . Finally, we

pressed as some quantiles of a symmetrized version of the d.f.

establish the most robust functionals with respect to these deviations. It is worth noting that these functionals coincide for both deviations.

2. RESULTS FOR GENERAL DEVIATIONS. Let

be an arbitrary deviation of the

family .;: and suppose that all assumptions (A-1) , (A-2), .•. , (A-b) are satisfied. Putting 1

.

- -

C ( K) : = ;- min (at ( K,£ ) - ao ( 1(, c), a, ( 11, c) - ao ( 11, c)) where we have written 0(11,£) for

fj(f(,c,1)

we may formulate the result concerning

the minimum risks.

THEOREM 2.1. (a) The minimum risk

t e, He) = sup

{ O( K) : K

4(£,1) (F)}

is given by .

(b) Suppose that there exists some funet ional Tc minimum risk

4 (e, 1-)

J

such that

T ( F)

=0

. Then the

is well defined and we have

B, (£·1)= max ( Bo (c .1

) , mine '!lax J =0,1

I (y F, £)),

r;'a.r I aj (,;;£)/))

J=O,1

For the proof of Theorem 2.1 we refer to Riedel (1985). The same method as used in Riedel

(1985) gives the minimum r i sks in the case where the functionals

considered

only

are

translation

equivariant.

Zi e.l i nak i (1985) have obtained the minimum risk

for general d. f. 's

the

.B, (£,1-)

latter

case

and

for unimodal d. f. 's F

• Applications of Theorem 2.1 to gross error devia-

and the gross error model tions

In

F

and symmetric ones are studied in Riedel

(1985).

COROLLARY 2.1. Let F be a symmetric d.f. (a) Then

(b) The functional

fa

defined by

is most robust.

3.MINIMUH RISKS OF DEVIATIONS BASED ON METRICS. Consider a deviation which is given by the

e

-neighbourhood with respect to the topology induced by a proba-

138

d

bility on metric

on

He ( K) =: {

II : L

: d ( K, L)

e ;u

E }

£,

For this deviation the assumpt ions (A-1), Moreover, if the metric

(A-2),

(A-3) are obviously satisfied.

has the properties

for all rf G!R and

d ( i,l)

(3.2)

= d ( K, L) , K, L £ I' '

then the assumptions (A-4)

and

(A-5)

are also satisfied.

variation distance, the Kolmogorov metric

Note that

,the Levy metric

the total

, the bounded

Lipschitz metric and the Prohorov metric are examples of metrics for which the corresponding deviations satisfy the assumptions (A-1), (A-2), ... ,(A-6) First we start with an upper bound of the minimum rislts.

THEOREM 3.1. The following inequality holds:

4 (E,1)

(3.3)

PROll':

Choose

{ Kn

sequence

a

lim

(3.4)

a,( F, 2 E , He )

;

n.."oo

C(Kn )

t

of

Kn e

d. f.' s

1: (F)

such

that

= BT (£,1>

and put

s.:

a·(K.n , £ ) and

In

for

.i »

J

t.In

:a.(kn , £ ) J

0 ,1. For analogy, we restrict ourselves to the case where

finite for all numbers (3.5)

n . Then (Kn )_ s. e He ( F) ;;n

and

are

;this is equivalent to

d(F,(Knts. ) e e . In

Hence (3.6)

Similarly,

it

follows

(3.7)

d (If

In

_t

On

from

,F)

£,

(3.1),

2 e .

(3.2)

and

the

triangle

inequality

that

139

From (3.6) and (3.7) we get

=f

C(Kn )

min

sOn ,tIn - tOn ):6

(SIn

-2'

a, ( F, 2

)

and in view of Theorem 2.1 and (3.4) the proof of the statement (3.3) is complete. It is interesting to

when does the inequality (3.3) become an equality.

Let us first introduce the following assumption on d (A-7)

For

any

two

d.f.'s

K

L

and

there

is

a

G

d.f.

such

that

d ( K,Ll ma.x(d(K,B),d(L,G)):6 - 2 -

(3.8)

Using the triangle inequality we see that that

:

K

(A-7) holds if for any d. f. 's

in fact,

an equality. Note

L

dCK,L)

/(fL

2

d(K'-2-)

(3.9)

and

(3.8) is,

is true. The assumption (3.9) holds for the total variation distance, the Kolmogorov metric, the bounded Lipschitz metric and for many others. However, for the Levy metric (3.9) is not satisfied. To see this, choose /(:=

0
c{ (I(,L)

K

and

L

with

+

let us define the d. f.

%rt ):=

(ma.r(/((t-E), L(f-e»+ min (/(Cf+c) , L t t» E))

From the definit ion of the Levy metric we conclude that L (i-E)-E :6 K(t+ E) -I- e As this inequality is also true for K:=:L tf

we get

.

max(KCt-E),Lft-E))-c

min (/((ttE),LdfEJ)+c and consequently

(3.10)

Using now the fact that implies (A-7) since

B£

converges weak Iy to

metrizes the

&;'0

(3.10)

as

convergence.

Unfortunately, neither (3.9) nor (A-7) are fulfilled for the Prohorov metric. For this fact we give an indirect proof and assume that (A-7) is true. Then, in particular, for /(:

= 80

and L:

{Wi th 0< E 0 . This inequality means that F is concave for t >0 and the sym­

l/

F

is convex for t '" 0

F

yields that

­

; i ;e , F

is an unimodal d. f.

For the proof of the sufficiency we assume that F is symmetric and unimodal. Then (3.19) is again valid and entails that F=F

because for symmetric d.f.'s

supremum in the definition of F can be restricted to

F the

0 . In view of Corollary

{j

3.2, (3.18) holds. It is worth noting that for an arbitrary symmetric and unimodal d.f.

Eo ( e , dr-v)

establ ish that Let us

£:

Eo ( e,

However,

in the general case we can only

dl() •

now turn to the problem of finding

THEOREM 3.5.

Let

F

the

dr v or on the Kolmogorov

deviation based either on the total variation distance metric has the same minimum risk.

F

be a symmetr i c d. f.

the most

robust functionals.

Then the most robust functionals

relevant to the deviation based on the Levy metric and on the Kolmogorov metric, respectively , coincide and the common most robust functional is given by T(K)= -;- CSl/p{

t- !1,( F, I((f)

+ C' J :

t < ()O( 1(, I­E) }

+ inf{t-QoCF,K(f)-eJ:t>a,(K, J}J. PROOF: In view of Corollary 3.1 the most robust functional with respect to the devi-

ation based on 1'( K)

is given by

=

ao ( 1(, e ,

­l­

2

a, (/c.s ,

)

Therefore, we have to derive S(K,e, df,) ( j

= 1,0)

when these expressions are

finite. By assumption, there is some SE/R such that ­ 00

« ao ( /(, e , c{)

£: S

at ( /(, s )


O,k= 1,2, is bounded. Thus we see from k , , H the above proof that LI, ' Llz are bounded in +' 1. Clearly, we may assume

U;

02 .

Now we combine (3.7') and (3.13) to ob-

tain first

Thus

I I ' =0 /..J,

by L.2.2. Now the same relations imply Li" z(tS)=OCexP(-'l 222 Sf )+ 8X,) whence

Ll; =0 follows from L. 2.2. 2.

Let

In

this

case

rewrite

(3.7)

in

the

form whence

follows. Sin"e we already I0 there

x(Ll) e [0, Ll J

exists ing

on

the

such that the funet ion

interval (-

00 ,

XC Ll »

FrXl-F(X-Ll) is non-decreas-

and non-increasing on [x(Ll) ,+ oo } •

PROOF. By the unimodality of F,

.f'or :r < 0

.for The function F(x)-F(X-Ll) has no negative jump at

(0, Ll)

,and has no positive jump at

Ll . •

X

> Ll

0 , is concave on the interval

160 1

F be a unimodal cdf with the mode at 0, let £ et t), '2), and let

L£tI4I\ 2. let (2.8)

Ll

»; Svp{ Ll : sup (FeX)- Fux- Ll») X

6 -E, }

-E

Then (1)

"*= 0

Ll

P=

iff

Z

(ii) There exists

T

(2.9)

z

,If.

F ( OJ - F ( O- )

Z

p

In particular, if

_c_ 1-

PROtF. (1) L)

*" is

e

«! - e) F + e) ::: Ll *" .

then (2.9) holds iff

c

U-E)F(O-)+

(2.10)

1-

EO [E,!-EJ such that

(i ]: E) F) - T "

ff.

=::

e

==

(I-£)F(O)

Z7f

*"=0

a finite non-neqat ive number. Obviously, Ll

¢;;f; 17 {.x: FrxJ-Fex-Ll) > ---.L} {.x:

.

d>O

f-E

The only possible jump of F is (ii) First suppose that

P

p


0

:!f. such that F( x)- F(x - L'.J)

O.

and, according to lemma 1, we can choose

is non-increasing for X

=

X (L'.J 'f). It is easy

to checl00.

Ll* and Z

such

additional condition liminf'liiCn-fL(n)-Z"»Uor

n

liminf C

>0

n

res-

pectively. We claim that

71'*=(%L Z

(n):n

-TJI'(F))co Z n= 1

is uniformly asymptotically most bias-robust statistic in the class

Ji . The state-

ment is based upon the following theorem.

ZE"Cc,r-eJ

THEOREM 2. If for any fixed

(L (T7))

n": 1

the sequence of integers

satisfies the conditions

,

(i)

=0

(3.1 ) (Li )

lim (n-fL(nJ-Z)

(3.2) (iii)

liminl n

(3.3) (iv)

liminfVii(1-c-n-'L(nJ»0,

J

Vii (n-fU ni- c) > 0

J

T1

then

lim n...,oo

PROOf.

The

(3.5) 13

and (3.6)

c)

r;«(1-£)F)-

(3.4)

idea

1XL cm :n

of

the

proof

as

follows.

7' «(;')-o} < 2' Z

(Vv> 0) (3!1o(oJ) (lin >

p{x G t tn y r n

is

VG'e TlCO))

Z

This implies that (Vn > maxi no (oJ ,

(O)}) (V G' e Tl({lJ)

We

show

that

164

7'z ((])- 0

/;f (

IJ, XL

tn ri n

)

T (IJ) ... Z

rJ

cn i: n

and hence

)

I-z{U- JF)+J

which proves (3.4). Taite

cS > 0

if = Cfr;

and define

C(

p{](X . == L (n).n 1ft]

is such that

k=Lrn)

fir;

(X /] Lt m t n

('k)7kU-Q)n-k.

, then

7: (B)-O)

p

(/] )-rJ) • Then

Z

,f:

rt

k=:L(n)

(kn ) e

k

n k

(/-C) -

and according to the Berry-Essen theorem

PIi(X l

(3,7)

L.

where

t/J

(ri ) :

tjJ (

n

6I- (/]) - 0 } Z

2

c+(/-£J

nE-L(n) )

Vnett-e)

+

2

VE(f-E)

is the standard normalcdf and c is the universal constant (cf . e.g. Bhatta-

charya, Rao [1976], th.12.4). Taite an arbitrary a (O,liminfVii(n-'L(nJ-Z») . Since V!i(E-ti'L(nJ) I, Ilmtn n V2E(1-£)nloglogn n instead

of

(3.2)

and

(3.3)

(cf.

Feldman,

nt t-se) - Len)

V2 E (t -

Tucl I, 5).

REMARK 2. Actually our solution is optimal in a wider class of estimators, namely in the class of asymptotically invariant ones. Obviously, i f then our sequence is the best one in the class of all estimators.

P .=

e

-1-

-£

166

4. Asymptotically most bias-robust estimators for particular models. In this section we give simple applications of the general result of the previous parts of the paper. Namely, for some specified parent cdf jC we write down explicit formulas for the asymptotically most bias-robust estimators of location under £ -contamination of the model J={F(x-B):B R}. Once aqa i.n we put a special emphasis on the Huber's solution, which covers many very important cases when F is symmetric. Our results allow us to extend the class of solutions also in the symmetric case (see e.g. examples 1 and 2 below). Observe that in order to determine the optimal se-

.:r we have

quence 71''N E Z

to find the appropriate Z:N' E [E,

t-

EJ

• Then we calculate

and choose the sequence of integers L tn} =L * (n) , which fulZ fils the conditions (3.1)-(3.3). We add the index z* to L *(17) to stress the

the values

rZ if ( F)

Z

correspondence expressed by (3.1) -(3.3). In the examples we omit the calculations which are trivial, and confine ourselves to presentation of formulas for the optimal

z", I*(F)

and the measure Ll

z

if

B

or,,) of

the asymptotic bias-robustness.

£ Z"

EXAMPLE 1. Degenerate parent cdf. Let F(x) 1

CO,=)

order stat istics (X

)

00

L(I7): 17 17= 1

(X).

Then every sequence of

such that (l (17»);' 1 satisfies (3.2) and (3.3)

is absolutely asymptotically bias-robust estimator of location under

S -contami-

nation.

EXAMPLE 2. Uniform parent cdf. Consider the cdf

FrXJ={ Z

For every

Z

L(I7):n

_

Z

x

1

-Z+

E

B (71')--E

1

x+y

E [£,1-£J

F=(X

1 -

e

1

:x:
O} ,

J = 0, .. . ,M

by

Let us denote

REMARK. The condition C1 trivially holds for N= 1 or

N > 2

Denote

are independent and they

Jk

}k ?: 1

-I

P( (N-I) 0901 < Sjr < ( N- t )

in the case

notation.

a sequence of i.i.d.r.v.-s on the probability space

are not dependent on the sequence

C1.

we introduce

j

M . Since the r , v .-s

are independent, from the condi tion C2 follows

O«N-lf'xo"" ll'...,;;

J

a:;.fr*«N-J)X*':

(N-l)

(N-1) S,,) > 0 S01)

:>

generation points must be considered on the 1-st queue.

0

, because an

. In this case the re-

185

=/'.;.(1),

Let us denote

(4)

{ k :

j == 0, ... ,,H M

-I

(N-tJl'f

J

=2

J

and K= N

.' s

I=R(k)-1(+1

R(k)-1

B.

t > 1'

!=Rrkj-N+1

and let us define recurrently the sequence of r , v , -s

where [xJ +==

max (x, 0) .

It is clear that the r.v.-s

of virtual waiting times in the inter-arr i val and serv ice times

YI(.== min

{

k : C* k

B

k,1

}

BIG)!

f R} k,1

f Ck if}k * Ck,k

.S , k I

f

1 , as follows:

mean the sequence

system with the sequence of independent

{(]k } If

c":» k k,1

Let us denote holds for eachk;;'J ,

186

then we have Y*

=00.

*

Let us define the events

fk

is easy to see that if the event

k::2

plementary part of the event

c

(

c

n ... n Fk

If

P ([e1

::::

.o"I J n [B',2 > eR

It is clear that

c

,

P(v=k)

c

where F: denotes the comk

. It is simple to prove that

C

(1 ) J

C

n fj; n '" n

* t

',I

n

In...nCCkIf-1 >.B.k -1

If

k

k,l

On the other hand the sequences

,{

K Ck+l- C k+1

the equal i. t y

notation lIo = l , Uk ",P(v=kJ;:{=O, (7)

From (3)

I'j

(1')

= . Uk_ 1+ ••. -I-{-1 we get ["Ok < E4,1 •

8' , J

assume

M the d. f. 1

SOM

that

the

fj IX)

condition

C2

is

satisfied.

is not degenerative, then from the

follows that there exists a number Ll > 0

+ Ll ) + P(

$'1). So J 1

+ Ll)J >0

•

,for

188

Without loss of generality we may assume that

P ( SOt::>

S"

+ L1 ) + P ( Sf! > SOl + Ll ) > 0

P( So, > SIt + Ll)

We prove the theorem in the case

:> 0

. I f the condition

P(Stt':>

So,+,1»O

holds, then we can prove the theorem in a similar way. ( In this case we examine the

sequence

of

regenerative

moments

of

time

in

the

1-st

queue

).

Denote

Let us define the sequence of r , v. . (11) the re.Iat ion (/(- 1M 2

*

'Jt1?

?:(IV-t):ro >(N-t):J:,j ;

=([

(N-fl L1

J> t, ... ,M

RW), R (f)

IS

»:0 ] + 1).:1

according to (5). From (2) and

follows, therefore in the

moment of time

T( R (kl)-eR(kJ the customers may be only in the CQ and the 1-st queue. Those customers which are in the moment of time T(R(kJ)-eR(k) , in the 1-st queue ,arrived at the queue after the moment of time i(R(kJ)- eR(k)

IV-I

the first queue is not greater than

• The number of customers in

and the sum of required service time is

not greater than

R(k)-'

r: =.;;E:7, k

denote

l:::

V

S

I, == max (R(k)-K+ 1, R(k)-N+ 1)

,where

.l

=mrn fl: Fl

Since the sequences { fk }k

eRa) }

?: f

.Let us and

f

are independent and on the basis of theorem 3 [7, p.199) they do not depend on the sequence {R( k) } and

PCv=k)= P( k

where

, then the r , v . V does not depend on the sequence

r, > eR(f) , ... , fk _t > eR(k_O •

eR(k)=

a- P (ifi (Soz- S,z) > k!J ) > 0 .

From (6) follows that there exists some

> 0 ,for which

member

('-(J)Eexp(4o lf )< ! ' Thus

£exppoR(v)} =£exp {ito co

k

k=f

i=!

'=.L': c J('i=k) exp{

i/

1::::f

1

r.}

0 1

fl eso k == 11-1 i5[U-I))(Eexp{Ao '1 }J] < from which the theorem follows.

OQ

,

{R(k ) J f1 ,

189

REFERENCES

1. Sauer C.H., Chandy K.M. Computer systems performance modelling, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1981. 2. Iglehart D.l., Shedler G.S. Regenerative simulation of response times in network of queues, lecture Notes in Control and Information Sciences, SpringerVerlag, New-York, 1980, 26. 3. Kalashnikov V.V., Qualitative analysis of the behaviour of complex systems by a

trial

functions

method

in

Russian

),

Nauka,

Moscow,

1978.

4. Szeidl l. On the estimation of moments of regenerative cycles in a closed Central-server queueing network ( in Russian ), Teorija Verojatn. i Primenen. (in print ). 5. Feller W. An introduction to probability theory and its applications, John Wiley, New York, 1970. 6. Szeidll.On the estimation of rate of convergence in the renewal theorem for a discrete irreversible case, ( in Russian ), Stability Problems for Stochastic Models. Proceedings of the Seminar,

Moscow, The Institute for Systems Studies,

1985, 121-126. 7. Borovkov A.A.

Probability

theory

in

Russian

),

Nauka,

Moscow,

l.Szeidl, Eotvos lorand University Computing Centre' Budapest, H-1117 Bogdanfy u.10/b

Received: May 10th,1985

1978.

ON F-PROC£SS£S AN> THEIR APPLICATIONS

L. Vostri"ova

ABSTRACT.We consider the f-divergences of probability measures on filtered spaces and define the corresponding f-processes. how the f -processes can perties

of

probability

We show

be used for the investigation of the promeasures

and

statistical

estimators.

KEYWORDS: f-divergence, Hellinger distance, Matusita distance, f -process, Hellinger process, Matusita process,convergence in variation, entire asymptotic separation,(Cn)-consistency. INTRODUCTION. The investigation of such properties of probability measures as absolute continuity and singularity, contiguity and entire asymptotic separation has brought the notion of Hellinger process

liptser and Shiryayev [10],

Memin and Shiryayev [11], Jacod [4], which seems to be useful for obtaining simple "predictable" criteria in the above problems. The Hellinger process is also useful in many statistical problems, for example

in studying the

(ell)

-consistency of estimators,

in obtaining limit

theorems for the likelihood ratio processes and statistical estimators etc. In these problems the general statistical parametric models are

usually considered

and the conditions on the Hellinger process becomes restrictive when the dimensionality of the parametric space grows. This makes it necessary to generalize the notion of Hellinger process, more precisely to investiagate f-diveregences and f-processes. This paper consists of three parts: in the first one we introduce the notion of the f -diveregences, in the second part we investigate the f -processes and the third part contains some applications.

191

lor-DIFFERENCE

OF TWO PROBABILI TV MEASURES. Let (Sl.,F) be a measurable space

P

with two probability measures P and

a .Denote

P« Q ,P«

= dP/dlJ and

by

=0

0/0

Letj=.!(x,Y)

and

¢ =dP IdQ

P with

Radon-Nil0

every

be a nonnegative homogeneous convex finite function on R x R

OCF INITION 1.1. (Csiszar [3]) The

.f -divergence of two probability measures

P

and P is

.P; ( P, P) : : :.

£Q.f ( ( ,

where £61 is the expectation with respect to

61 .

REMARK 1.1. From the homogeneity of .f we get that the j -divergence does not

a and

depend on the dominating measure () . Indeed, i f Q« Z'= dQ/d{)'

dP/dQ',?'=dPldf:J.:

then

In the general case we use these equalities for the pairs of measures Q, and Q', ( Q + Q')

12 .

REMARK 1.2. The f-diveregence of P and true for Q = (P + P) /2

measure

Q'

P t akes only finite values. This is

because of the continuity of convex finite functions on

open set and the inequalities 0 dominating

(61 + Q')12

we

=

0 }nF= r, the

Lebesque monotone convergence theorem.

COROLlARY2.2.For every stopping time

.P U;., fiT ) ==..P/ f

T

T we have

! 9 ( I, Zs-) dhe

+ E 0

result follows from the

196

IT = PI Fr '

where

= PI

.

let .¥ be the set of all

1F -stopping times and IfJ{Zrl=Rf,Z7:) -:I CO, 1)21: for

gO,zsl,C2 =SlIp{£/ rp(Zrl!r:t:/kx.

to,]'. For a given a>! we set

s>O 'l" ,J: there isa>!thatq.O we

THEOREM 2.2. have

_

Pa) + c, E + 2 c2 P

(P, p ) ..p;/ (

jJ

where

=IX /

(

ex -

flfJ

( hco

?:

8)

1) •

.Since h is a predictable process,

PROOf. Let

re

is a

predictable stopping time and there is a sequence of stopping times (z,/lk 1 such k

that I" < r • r '"

II

k

as k-+oo.

t7:

II

For every

and

t >0 we

get

;

,15 k )+,?,(P k 'eAt f 't:'e Af

reAt

f

,Prk/l t e/l

)

and by corollary 2.2 (8)

e

+ Cf

e

8

'

,

the process h is being nondecreasing. Since.p, (. , .) does not depend on dominating measure we can t ake 11

f

=(P+ PJI/?

Changing the measure ( see Liptser,Shiryayev [10] we have

= E:I(f,Zt)+f(O,n p (

l=

(9)

't

=0)=

«(l,IJ

By (9) and the Cauchy-Schwartz inequality we obtain -

P

f

(10 )

:f

E I

0

198

n n tx n wherejJe is the Hellinger process corresponding to P, P and

IF

n

.By theorem

3 of (12] we also have

n

n

2

e)

P (dt'oo

E + (f + 2/ e

) .fJ2 (P n, P-n)

2

,

which together with (11) gives the result. REMARK 3.1. In discrete time this result was proved in Vos t r Ikova [15], for point processes in Liese [6] and in the general case it was obtained by Liptser and Shiryayev [10] and Kabanov[5].

n -/? DEF INITION 3.2. (Le Cam [7]). The sequence (P, P )

is called entirely

n

asymptotically separable ( denoted (P ) Ll (P

) ) if there exists a sequence . . . n' n' I and P (A )-+1 as n-+oo.

n r n' n' n' A e F such that P (A Let

n

e" and

be the Radon-Nikcdym derivatives

n -n (P + P ) /2

to

Q::::

for

a >0.

defined as in (1) and

Z/ Z:!

We also set (Xn=

PROPOSITION 3.2.

lim

n

-n

I {

Z =

wi th (;(on=

of

P

n

n

with respect

supposing OIO=o,aIO=oo

1.

Suppose that

lim pn( inf ex; S >0

1/ R)

n-,>oo

O.

n

_n

for the entire asymptotic separation (P ') Ll (P)

Then

-rn

P

and

a necessary and

sufficient condition is

lim

L.,.oo

lim Clo

-n ri c-r: n xn ,r:r;::::nPROOf. Notice that (P)L1(P ) if lim f{(P,P )=0 where f{(P,P )=EnVSS"

n

rr-o cx:

is the Hellinger integral of

1/2.

oe

0, = (JRL)

REMARK 3.2. Liptser,

-1

theorem £:;

4

of

,c2 = ex p ( c,-2+ 2)

(13]

we

get

2L)}'/2

n( inl cr n C, + C f1 ( p7 pn) + p 2 8>0 t

L)

pn(/}fn

{J

From

H(P7p,)

(12 )

where

order

fiR)

,which proves the proposition.

The similar result in discrete time has been obtained by

Pul

•

Thus the sequence (C ) is such that the distributions of random variables

n

CnlBn - a')

with respect to Pen, are tight uniformly over the set

and it also has the maximum order of magnitude. In order to formulate the results we introduce some notations. We say that a nonnegative function 0.

(15)

where JJ' is a constant independent of

L.

PROOf. Using theorem 1 of [14] wi th r-emarks 4 and

q

-

lim

1)

Sl/P

n.o,oo

2)

lim

Wherep

q

n

fltj (PO

sup

n

I

PO+C-f l/ 13 n

luI

IO/£:L. IU/6L.

n

n

) 6 IlL


N)

and mave use of inequality (12). Let us go over to the proof of point 2°, i. e. to the proof of

02 c -indepen.

2 dence statistics X and SN • It is not difficult to get convinced that N

(13)

$;

u,


0 and A is defined on the

CJ ( £) 2': C/a ( E)

Thus, let us choose a variable

(see (32),

-1

(n-1)

N (t .. nIt j

where fJo==goCc)=V2Alnl/E basis of inequality

4

. Making use of (30), we obtain:

/RitJj

Repeating

IR (t)/

R ( t) with 4

•

A .

Since for

It I

q

221

2

(41)

Let us get convinced that if lJA=

eO i.e.

,then for

V(3ln 1/£)/(]](n+2)) .

that This

n- no 8/2

is

carried

If(tJl

out

by

a

direct

exp(-17Aln f/c) 4£

For the given value lJA:=:

substitution

of

(39)

N

(102 + nqo3 /8)

!-no- 0/2 the right-hand part of this inequality is not

constant, depending only on

n.

X1

where

ep )

Substituting the value

C5 ( Zn 11£)

1J

q/=

is some positive

1/£

F;(X)=P(Z