Moments of probability distributions semiattracted to semi-stable m easures on Hilbert spaces

A (' T A U N I V F R S I T A T I S L 0 D Z I E N S I S FOLIA M AT HE M AT ICA 9, 1997 M aria Kłosowska M O M E N T S O F P R O B A B I L I T Y D I S T R I B U T I O N S S E M I - A T T R A C T E D T O S E M I - S T A B L E M E A S U R E S O N H IL B E R T S P A C E S 1

Let. H be a real s e p ar a b le Hilbert, s p a c e , </ a lio n - d e g e n e r a t e s e m i-s t a b le d ii-s t r ib u t io n on H an d « £ ( 0 , 2 ] an e x p o n e n t for q.

It. i.s pr ov ed t h a t t h e p r o ba b il it y d is t r ib u t io n s s e m i- a t t r a c t e d t o t he m e a s u r e r/ h a v e a b s o l u t e m o m e n t s o f order / i for ß £ ( 0 , o ) an d h av e no s uc h m o m e n t s for /i > n and о ф ‘2.

Let II be a. real separable Hilbert space with the norm | * |. Consider

where .V, are independent //- v a lu e d random variables with a com m on distribution />. a n > I), x n 6 II and { kj } is an increasing sequence of p o sitiv e integers such that

(2) lim kn + 1 k u 1 = r < -boo.

T he distributions o f sum s ( I) m ay be written in the form

where the power / / is taken in the sense o f convo lutio n, Sr denotes the distribution concentra ted at a point x G / / , and the m easure T ap is defined by the form ula

for all Borel sub sets В of / / .

A probability m easure ou II is said to be se m i-sta b le il it is a weak lim it o f sequence (3). VV.M. Kruglov g av e in [4] a characterization of se m i-sta b le m easures. Nam ely, a m easure on II is sem i-sta ble il and only if it is a G aussian m easure or an infinitely divisible purely Poisso- nian m easure represented by a L évy-K hint.ehine spectral m easure M such that

for som e n G (0, 2) and A G (0, + o o ) \ {1}.

T he class o f sem i -stable m easures is a subclass o f infinitely divisible m easures and is a natural ex tensio n o f the class of sta ble m easures. For this reason, in the sequel, the num ber о in (4) will be called an expo nen t for a purely P oisso nian sem i-sta b le m easure (the expo nent for a G aussian m easures is equal to 2). S em i-sta b le m easures have their dom ains of se m i-a ttr a ctio n . Nam ely, by a d o m a in o f s e m i- a t tra c tio n o f a se m i-sta b le m easure q we m ean a class of distributions p such that sequence (3) converges weakly to q for som e a n > 0, x n G I I and { A‘n} sa tisfy ing (2). We shall also say tha t p is se m i- a t tra c te d to q if p belongs to this class.

T he theorem s on m om ents of m easures a ttra cted to sta ble laws can be found in [1] and [Г>]. We shall prove an analogous theorem for

distributions sem i-a ttra c ted to sem i-sta b le m easures on I I . Our proof is elem enta ry in the case r > I and. in ih e case /• = 1 (th e stable case), we can apply th e sam e m ethod. Consequently, if we reduce the problem to m easures attra cted to stable laws on a straight line, then we obtain a proof sim pler than the classical one.

T h e o r e m . Let q h e a n o n - d e g e n e r a te s e m i - s t a b le m ea su re on H, a n d n G (0.2] an e x p o n e n t for q. I f a d istrib u tio n p on II is s e m i - a ttr a c te d to th e m e a su re q, th en

Tup ( B ) = p { x G II : a x G В },

a n d

/ |.r|',,/)(r/;r) = -l-oo, for ft > O', О Ф 2.

« II

Proof. W e shall consider several cases.

C A SE I. Let n G ( 0 , 2 ) . Thus the m easure q is represented b y a

L évy -K h in tchin e m easure M ф (J. From the assum ption we can find sequences {« „ }, {•*'„}, { k n } such that

Using Corollary in [2] and Lemm a 7.1 in [6], we obtain th e following: for any £ > 0, th e sequence of m easures {h nT n- i p } restricted to the set

G II : M > £}

is weakly convergent to the m easure M restricted to the sam e set. In particular, we have, for som e t. > I),

(6) litu /.•„/'{•'■ G II '■ И > /« Л = Л /{.r G II : |.г| > /} > 0. Ca s e I A . Let T hen and (s e e [4]). Of c o u r se , lim A: „ +1 k 1 = r > 1. »I —'X) li m

_—

_iVl

and, since a < I, we ran assum e t hat {«„} is an increasing sequence. By (()), pu tting bn = t a H, we have

,• /»{•'■ G I I : И > I>„+1} -1

( hin — :— — —— r—,--- j—j - = Inn А;,Д-1)+1 = r . ,,- x , /){.!• g / / : |.»i > />„} '

Let us now consider a series o f the form

È

e

By (7) and d A lem b ert’s Criterion, we obtain the convergence of the series il и V-1 < 1, i.e, j( ß £ (0, о ) , and the divergence if fj > o . It now suffices to m ake use o f the inequalities

(8) j M V r f * ) = E J \*\fii № ) ,/M>*»! n=l Л„<|а?|<6и+| < 6 II : 1*1 > M «=1 Un and (0) / /»(d*) > ż * ü k * e w : 1*1 > - r i r 6 //. ; ' T ' , > V : >) . «=1 v 1>{-r e II : |.r| > /)„} / Ca s e IB . Let Jim /.„+ i/ .- 1 = г = I, Then lim = « = I

and q is a sta ble m easure (see [ I] ). Consequently, the m easure M has the property

7 \ Л / = А " Л / ,

for each A > 0.

T hus, for all t > 0, the set

{ r € / / : | z | > / }

is a continu ity set of M and condition (6) is satisfied for each t. > 0. T his im plies that

— Л

for each t > 0, and, since a = 1, we obtain p { x e I I : \.r\ > t u ) (1 0) hm — 7--- —— Г--- г = t

«• 7>{;r G II : |.x| > (/) for each / > 0.

By putting t = 2 and « = 2n in (1 0), we have £ / / : H > 2"+' )

e II : | r| > 2"1 ‘ ' T hus, the series of the form

Ê ( 2“ ) M * € / / : M > 2“ } n=i

is convergent for ß E (0, <v) and is divergent for ß > a . It now suffices to use inequalities (8) and (9) for bn — 2” .

Ca s e II. Let о = 2. Ill this case we can assum e that

I |ar|2;j (rfa-) = + 0 0.

•J //

Now, <i is a G aussian m easure on II represented by a non—neg ative, self-a d jo int operator .S' with a positiv e finite trace. Let {a ,,} , {;cn}, { k n } be sequences such tha t the condition o f form (5) is satisfied. In the sam e way as in the proof of Theorem 3.2 in [3], condition (5) im plies now

(1 1) lim kna~2 f \.r\2/>((!,r) = t v S > 0 n _ v

for each I > 0.

CASE IIA . Let r > I. Then a G (0, 1) and u.2r = 1 (see [4]). We have

lim «„ = -boo 71 — OO

and, since a < 1, we can assum e that { a n } is an increasing sequence. For ß € ( 0 ,2 ), we have th e following inequality:

(1 2) I M 'M A r ) < £ I |.r|V (< b). By using (11) for t — 1, we obtain

lim Iin, K < 1

4 i<„,. M W* 0 — » " +1

= a ~0r ~ l < a ~2r ~ x = 1. T hus the series in (12) is convergent and

f n M 2P{<1*) < + 0 0

for ß G ( 0 ,2 ) .

C a s e H B . Lei / • = 1. Condition ( II) im plies that

lim = 1

'l~ 4 / M<„„ M W * ) lor each / > 0 and, since <7 = 1, we further have

( i3) [im jk<«» m v w = t

/w<„

for each / > 0.

P uttin g 1 = 2 and и = 2" in (1 3 ), we o btain, for ß G ( 0 ,2 ),

lim — - ]1''1<2;,+1 |-r|2/,(f/'r) = 2fi~2 < 1 / |г |< 3 » M W * )

T he ab ov e inequality m eans that th e series

Е (2' Г - 2 / . M W - ' -) „ = 1 J\r\<i”

I

is convergent,. It now suffices to m ake use of inequality ( 12) by puttin g a — ‘>n

u1l — — •

Our theorem im plies the following

Corollary.

R e f e h e n o e s

[1]. В. V . G n e d e n k o , A .N . K o lm o g o r o v , L i m i t D i s t r i b u t i o n s f o r S u m s o f I n d e p e n

-d e n t R a n -d o m Var iables , C a m b r i-d g e , 1954.

[2]. H .J a j t e , O u c o n v e r g e n c e o f i nf i ni te l y div isi ble d i s tr i b u t i o n s in a Hi lbe r t s p ar e, C o llo q . M a t h . 1 9 ( 19 68 ), 3 2 7 - 3 3 2 .

[3]. M . K ło s ow s ka , T h e d o m a i n o f a t t r a c t io n o f a n o r m a l d i s t r i b u t i o n in a Hilb er t

s pac e, S t u d i a M a t h , 4 3 ( 1 9 7 2 ) , 1 9 5 - 2 0 8 .

[4]. V . M . K r ug lo v , On a cl as s o f l i m i t lams in a Hilb er t s pa ce , Lit.. Mat. S b 1 2 ( 1 9 7 2 ) , 8 5 - 8 8 .

[5]. II .G .T uc ke r, On m o m e n t s o f di s tr ib u t i on f u n c t i o n s a t t r a c t e d to s t ab le laws . H o u s to n .). Mat Ii. 1 ( 1 9 7 5 ) , 1 4 9 - 1 5 2 .

[6]. S .K . S .V a r a d li a n , L im i t t he o r e m s f o r s u m s oj i nd e p e n d e n t r a n d o m v a r i ab le s

uiilli v a l a i s in a ll il hc r l s pace , Indian J. S l a t . 2 4 ( 1 9 0 2 ) , 2 1 3 - 2 3 8 .

A / a r i a K ln. se ,w s k n

M O M E N T Y R O Z K Ł A D Ó W P Ó L P R Z Y C I Ą G A N Y C H P R Z E Z M I A R Y P Ó L S T A B I L N E

W P R Z E S T R Z E N I H IL B E R T A

Niech II będzie rzeczyw istą, ośrodkową przestrzenią H il berta, q - niezdegenerowanym pólstabilnym rozkładem pra wdopodobieństw a na / / , a <v (E (0,2] - wykładnikiem rozkładu q . W pracy udowodniono, że rozkład pra wdo podobieństw a na / / półprzyciągany przez q m a m

o-m enty absolutne rzędu ß dla fi e (O .n) i nie m a takich m om entów dla

ß > n i n ф 2.

lnst.it.ute o f M a t h e m a t i c s U n iv e r s it y o f Łód ź ul. B a n a c h a 22 . 90 - 23 8 Łód ź , Po la n d