Domain of operator attraction of a Gaussian measure in RN

ANNALES SOCIETATIS MATHEMATICAE POLON A E Series I: COM MENT ATIONES MATHEMATICAE XXII (1980) RO CZN IKI POLSK IEG O TOWARZYSTWA M ATEM ATYCZNEGO

Séria 1: PRACE MATEMATYCZNE XXII (1980)

M. Kl o s o w s k a (Lôdz)

Domain of operator attraction of a Gaussian measure in R N

Let {X n} be a sequence of J^-valued independent and identically distributed random variables. Consider the sums

(*) An( X l + X 2+. . . + X n) + bn,

where A„ are non-singular linear operators in RN, bne RN. The paper aims at characterizing a class of these distributions p of random variables X t for which at a suitable choice of norming operators An and vectors bn the sequence of distribution of random variables (*) is weakly convergent to some full Gaussian distribution in RN. The detailed description of the class is given for the case of positive and diagonal operators An.

Let be the set of all Borel probability measures on the real Euclidean space RN. being endowed with the topology of weak convergence of measures and the convolution as an operation, it constitues an Abelian topological semigroup. Convolution of two distributions p and q will be denoted as p* q. ôx will be a probability measure concentrated at a point

x eR n .

If A is a Borel mapping of RN into RN and р е Ш , then Ap is a measure defined as follows

Ap{E) = р ( А ~ 1Е) for every Borel set E in RN.

The characteristic function of a measure p will be denoted by p.

Evidently

Ap{y) = p( A*y) for every y e R N, where A* denotes the adjoint operator.

A measure p £ sJ01 is said to be full if its support is not contained in any (AT— l)-dimensional hyperplane of RN.

In [7] M. Sharpe introduced the notion of an operator-stable measure in R N. A measure q is said to be operator-stable if it is a weak limit of a sequence of measures of the form

(

1

where pe9Jl, A„e G, bne R N. Here the symbol G denotes the group of all non-singular linear operators in RN. The general form of characteristic functions of full operator-stable measures in RN is given in [4].

A domain of operator attraction of an operator-stable measure q is a class of these distributions p for which there are sequences Ane G, b„e RN such that the sequence of distributions (1) converges weakly to the distribution q.

In the paper we shall consider a class of distributions attracted in the operator sense by a full Gaussian measure in RN, i.e. by the measure whose characteristic function is of the form

q (y) = exp [i (x0 ,y ) - i ( S y , y)J for every y e R N,

where x0gRn , S is a linear self-adjoint positive operator acting in RN. S is a dispersion operator of the distribution q.

Lemma 1. I f lim A„p"* * Sb = q, where Ane G, b„e RN, p, q еШ and the

n-*ao n

distribution q is full, then the distribution p is also full.

The lemma is an immediate consequence of Lemma 2 in [3].

L^emma 2. Let A„e G, b„e RN, p, q еШ and q be a full Gaussian distribution with a dispersion operator S. The condition

(2) lim Апр* *ôb = q

п-+оо is equivalent to the following

(a) lim n f A„p(dx) — 0 for every e > 0,

" - >Q0 \ \ x \ \ > t

(b) lim n { f (x, y)2 A np ( d x ) - [ f {x, y) Anp{dx)]2} = {Sy, y)

IUÎKe llxlKt

for every e > 0 , y e R N.

P ro o f. By Lemma 1 in [2] condition (2) implies A„ -* в which means that the distributions A„p are uniformly asymptotically negligible. On the other hand the above property of the distributions A„p follows from (a).

Thus we can make use of Theorem 6.3, p. 200, in [6], and all we have to do is to show that (a) and (b) are equivalent to

(a^ lim J A„p(dx) = 0 for every e > 0,

(bi) lim n j ( x - x n, y)2 A„p(dx) = (Sy,y) for every e > 0 , y e R N,

"-*00 ||x|Ke

where

(3)

Gaussian measure 75

(*„,>0= j (x , y ) A np(dx).

IUIK1

Uniformly asymptotic negligibility of the distributions A„p guarantees that

(4) lim ||x j = 0.

n~+ oo

Thus (a) and (b) are equivalent. Simultaneously, for е е ( 0 , 1)

«I f ( x - x „ , y ) 2 Anp(dx)~ J (x, y)2 A„p(dx) + [ j (x, y) Anp(dx)]2\

H x l K t ||x||<e l l x ll ^ e

< 1М|и

J

J

||x||>e l l x l l ^ e I W I ^ e

Thus if (a) is satisfied, then (b) and ( b j are equivalent.

The symbol py will denote a distribution induced by an element y e R N, i.e.

(5) f { Z ) = p{x gRn : (x , y ) e Z } for every Borel set Z in R 1.

Lemma 3. I f a distribution р еШ is operator-attracted by a full Gaussian distribution qe$R, then for every 0 ф у e RN, the distribution py is attracted by a non-degenerate Gaussian distribution in R 1.

P roof. Let lim Anpn* *ôb = q. In terms of characteristic functions we

n -* 00 "

have

(6) lim Ip (A* y)Y el{bn,y) = e~i(Sy,y) for every y e R N.

и —* oo

Let 0 Ф y0e R N. Since A* e G, for every n there is 0 Ф y„e R N such that

^*(Уп) = Уо, i-e.

Уп Уп

Let now z0 Ф 0 be the limit point of the sequence---, i.e. z0 = li m ---—

II y j lly„J

and let t be an arbitrary fixed real number. The convergence in (6) is uniform in each bounded set and thus

We can take operators A„ such that S = I, where / is the identity operator

in RN. Then we have (Sz0, z 0) = (z0, z 0) = 1. Thus the distribution py° is attracted by a non-degenerate normal distribution in R * 1, and the numbers

UyJ-1 form a sequence of norming constants.

R e m a rk 1. The domain of operator attraction of a full Gaussian distribution in R N coincides with the domain of operator attraction of a Gaussian distribution with the identity dispersion operator /.

Le m m a 4. I f a distribution p is operator-attracted by a full Gaussian

distribution in R N, then there are a sequence Dn of positive diagonal operators, a sequence Un of orthogonal operators and a sequence bn e RN such that

lim Dn Unpn* *ôb = q, ft -*• ⁰⁰

where q is a Gaussian distribution with the identity disporsion operator.

P ro o f. By the assumption we have

(7) lim Anpn**ôc — q, where AneG, c„gR n.

n -*■ 00 ”

In view of Remark 1 we may assume that the distribution q has the dispersion operator I.

The operator Ane G may be written in the form An = V„Bn, where V„

is an orthogonal operator and Bn is a positive self-adjoint one. On the other hand B„ = U ~x Dn U„, where D„ is a positive diagonal operator and U„

is an orthogonal one. The sequence of operators Un V f 1 is compact. Thus if lim U V f 1 = l / 0, then

k~* oo к к

lim D U pnl* S b = U0q = q, where bn = Un V f xcn.

k~*oc K K nk

Thus the sequence of distributions Dn Unpn**ôbn is compact and all its convergent subsequences have the same limit q.

Let us assign to a full distribution р еШ and to an arbitrarily fixed basis [e1}...,e;v] in RN the following correlation matrices with elements of the form

(8) R u (a)

J xf Xj p (dx) — J XiP (dx) J Xj p (dx)

M(a) M(a) M(a)

j j x f p ( d x ) - [ J Xip(dx)]2}112 { f x j p ( d x ) - [ J Xjp(dx)]2}il2 ’

M(a) M(a) M(a) M(a)

N

where 0 ф a e R N, M(a) = \ x e R N: £ a f x f ^ 1], x t = (x ,^ ), at = (а,е{) i = 1

for i = 1 ,2 ,..., N, and functions of the form

(9) 0,-(a) = a2 [ J x 2pei( d x ) - \ j xpe4dx)]2},

at jjcf < 1 аг|дс| < 1

a > 0, i = 1 ,2 ,..., N.

Gaussian measure 77

R em ark 2. If the distribution pe‘ is attracted by a non-degenerate Gaussian distribution in RN, then

Th e o r e m. The full distribution р еШ is operator-attracted by a full Gaussian

distribution in RN with a sequence of positive diagonal in a fixed basis norming operators if and only if for some basis [_el , . . . , e n] in RN we have

i.e. the distributions pei, i = l , 2 , . . . , N , are attracted by a non-degenerate distribution in R 1 (see [1], Theorem 1, §34);

2° for i,j = 1 ,2 , . . . , N , i < j, there exists a limit o f the function R(J(a) as a tends to zero, so that

(See (8) and (9).)

P ro o f. Necessity. Let lim Dnpn* *Sbn = q and let [e t ,..., eN] be eigen­

vectors and [A*,...,A^] eigenvalues of the positive diagonal operators D„.

Condition 1° follows directly from Lemma 3.

The distribution q is a Gaussian one with an arbitrary dispersion oper­

ator S. Notice that without any loss of generality we may assume that (5с(-,с,) = 1, i = 1, 2, . . . , N. To this aim it suffices to consider a sequence of norming operators of the form D0 D„, where D0 is a diagonal with the eigenvalues l/yJ{Sei, et), i = 1 , 2 , . . . , N . From Lemma 2 for any e > 0 we

Since ||D„x|| = Y, therefore using (12) and (13) we obtain

(10) lim gfia) = 0.

have

(ID lim n j p{dx) = 0,

"-*00 \ \ D „x\ \ ^e

(12) lim n (Aj,)2 {* j x f p ( d x ) ~ [ { x,p(dx)]2} = 1, (13) lim nÀÎ,Ài[ f Xi Xj pi dx)- f x iP(dx) Г Xjp(dx)]

n->00 ||Г )„Х ||<£ ||Г>и*11^е

= (Sei, efi.

N

lim Rij(Xn) = (Seh efi, where A„ = У А1пе(.

i = 1 (14)

Since

(15) lim l p( Kei t ) Y ■ eit(bn,ei) = for every t e R 1

and lim AÎ = 0, i = 1 , 2 , N, we have by Theorem 2, § 10, in [1],

(16) lim

П-+СО A

al = 1, i = 1 , 2 ,. . .,1V.

n+ 1

Let be an arbitrary sequence tending to zero. From (16) it follows that there is a subsequence m(n) such that

(17) lim

l m (n )

= 1

To show condition 2°, the sequence {а^}®=1 need to be chosen so that

(18) ^{, •} 9 1 ( ^ n ) , • T 1 AT

lim ---- r— = 1, j — 2 , 3 ,..., IV.

9

Mi)

From (15) follows (see Theorem 2, §26 in [1]) (19)

9

Mi)

Thus in view of (19), (17) and (18)

lim д —~П:^ - 1, j = 2, 3 ,..., N .

0/ M m (n)) _ J’m 9 j (Ащ(п)) 91 (Am(n)) 9 l ( a n ) _ J 9 j ( a l ) n^°° 91 (Am(n>) 9i ( a l ) g j ( a i )

Since lim m (n) gj (A4(n)) = 1 we have

n-> 00

(20) lim m(n)gj(aJ„) = 1.

П GO

Condition Г and (20) imply

(21) lim m(n) J p(dx) == 0 .

«-► OO : a}n\xj\>e Thus for some sequence o ^ e # 1 we have

(22) lim [p (aJn e} t)]m(w) for every t e R 1.

n~> СЮ

In view of (15) and (22)

(23) lim AkïL = ! for j = 2, 3 ,.... N . a„

Finally by (14) and (23)

Gaussian measure 79

N

lim Rij(an) = [Sei t ej), where an = £ a'e,..

i=l

Sufficiency. From condition Г it follows that there are sequence of positive numbers {a|,}®=1, i = 1 and sequence b„eRN such that (24) lim p i a i e j t f elt{b"’ej) = e~il for every t e R 1.

Consider the sequence of distributions

(25) Anp * * S bn,

where A„'s are diagonal operators such that Ane} = aJnej. Condition (24) means that sequences of one-dimensional boundary distributions of the sequence (25) induced by the elements of the basis are weakly convergent to Gaussian distributions in R 1. Hence it follows that the sequence of distribution (25) is compact and that each subsequence which is convergent converges to a Gaussian distribution in RN. Thus let S t and S2 be two dispersion operators of limit Gaussian distributions.

By (24) we have (Sl ei, e i) — (S2^ , e,) = 1, i = 1,2, which further yields the existence of the limit

(26) limn(aj,)2 { J x f p ( d x ) —\ j Xip{dx) |2} = 1.

" ' * c0 lM „ j c | | < e \\Anx\\ $ e

By (24) the sequences i = l,2 ,...,i V , satisfy the condition

(27) lim

Л-+ 00 9 ifcn1)

= 1.

Thus condition 2° along with (27) and (26) guarantee the existence of the limit

lim nal„ aJ„ [ J x, Xj p (dx) — J x, p {dx) J Xj p (dx)]

”~+a0 1И„х||«г

for every £ > 0. Thus (Sr et, ej) = (S2 et, ej) for i , j = 1 , 2 , N, i.e. S x = S2.

Co r o l l a r y. A full distribution р е Ш is attracted in the ordinary sense

by a full Gaussian distribution in RN if and only if for some basis [_ег, eN~]

in Rn

X 2 p [x eR N : \Xj\ ^ X}

(i) lim

X-* + oo J x 2 p ( d x )

\xj\ZX

0 for j = 1 ,2 ,..., N;

(ii) there exists

J X i X j p ( d x ) — f X i p ( d x ) f X j p ( d x )

INI^x IMI^x IWKx

*'+™00 [ J x 2p ( d x ) - [ J Xip(dx)]2f { J x j p ( d x ) - [ f Xipidx)]2}*

llxll^X \ \ x \ \ z x IM K * \ \ x \ \ s x

for i , j = 1 , N , i < j;

(iii) there exists

j x 2 pei (dx)—[ j xpei (dx)]2 j x 2pei(dx)— [ j xpei(dxj]2 Ф 0 for i = 2 ,3 ,

Theorem in the paper describes the domain of operator attraction of a full Gaussian distribution provided that the sequence of norming operators is a sequence of positive diagonal operators. As seen from Lemma 4, the general case reduces to considering a sequence of norming operators of the form Dn Un, where Dn are positive diagonal operators and U„ are orthogonal ones. Unfortunately, this case is still the problem to be solved.

[1] В. V. G n e d e n k o , A. N. K o lm o g o r o v , Limit distributions for sums of independent random variables, Cambridge 1954.

[2] R. Jajte, Semi-stable probability measures on R*, Studia Math. 61 (1977), p. 29-39.

[3] J. K u c h a r c z a k , On operator-stable probability measures, Bull. Acad. Polon. Sci. 23 (5) (1975), p. 571-576.

[4] — Remarks an operator-stable measures, Colloq. Math. 34 (1) (1976), p. 109-119.

[5] P. L evy, Théorie de Paddition des variables aléatoires, Paris 1937.

[6] K. P. P a r th a sa r a th y , Probability measures on metric spaces, Academic Press, New York-London 1967.

[7] M. S h arp e, Operator-stable probability distributions on vector groups, Trans. Amer. Math.

Soc. (136) (1969), p. 51-65.

INSTITU TE O F M ATHEMATICS UNIVERSITY, EÔ D Z

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