infinitely divisible and stable laws in separable banach spaces. ii

INFINITELY DIVISIBLE AND STABLE LAWS IN

SEPARABLE BANACH SPACES. II

V. Paulauskas and A. Rackauskas UDC 519.21

The present paper is a continuation of a paper written by the first author, but since after the first part

[38] was written, the second author obtained results on domains of attraction in spaces including Lp spaces,

I -< p < 2, we decided to add these results to Sec. 4. Without recalling them, we shall use all the notation and

lists of literature given in the first part, so the list of literature at the end of the paper begins with number 29.

We note also that when the first part of the paper was in press, [43] appeared, in which, in addition to other

results, a result was given which coincides with Theorem 2 of the first part.

4. Domains of Attraction of Stable Laws in Banach

Spaces with Schauder Bases

Let ~l, 42 .... , ~n, �9 �9 �9 be a sequence of independent identically distributed (i. i. d.) B r. v. with distribu- n

I tion F. Let F n denote the distribution of the sum -~. ~ ~i-a,, b,>0, a~eB, normalized in some way and centered.

i--I

Letp be a stable distribution on B with exponent ~, 0 < o~ < 2. We say that F belongs to the domain of attrac-

tion of the distribution p if F ~ (~ denotes weak convergence of distributions). If b n = nl/~, then we say

that F belongs to the domain of normal attraction ofp. In the case of a Hi[bert space H one has a complete

description of domains of attraction of stable laws in H [16], but in the case of a B space even the domain of

attraction of a Gaussian law has not been described completely, and all the results on domains of attraction

of stable laws in B up to now are in the form of preprints [20, 17, 30], and some of the results given below are

taken from [20]. We shall consider Banach spaces with Schauder bases of S-type p, p -> i, "which were intro-

duced in the first part. (We recall that all B spaces with Schauder base are spaces of S-type i, Ip is of S-type

p, 1 _< p < ~o.)i Then to establish /:,,~ we must show that the finite-dimensional distributions (f. d.) Fn con-

verge to the corresponding f. d. p and the sequence of measures F n is conditionally compact (c. c.). First we

give a general remark on convergence of f.d. In the case of convergence to a Gaussian law, convergence of

f.d. is guaranteed by the condition Eli ~tll 2 < ~o but in the case of convergence to stable laws in B it is no longer

sufficient to require regular behavior of the probability P {II ~t II > r} as r ~ ~o, since simple examples in C ([0,

1]) show that f rom regular behavior of the probabili ty p{ sup t~ (t)I> r } it does not follow that P {I ~ (t0) ~>r}, O~t~<l

t o being some point of [0, 1], var ies regular ly .

Let ~k, k --- 1, be some family of pro jec tors B - - R k. Since we consider a B space with a Schauder base, one can always take the family ,Vk, introduced in Sec. 3, however one can also use other families of pro jec tors . For example, in C([0, 1]) the pro jec tors ,Vk, defined by the formula ,VkX = (x(t l) . . . . . X(tk)), x e C([0, 1]), tl, fe . . . . . t k being points of some countable set which is dense in [0, 1], a r e m o r e convenient. By ,VkF we shall denote the distr ibution in R k defined by the formula nkF(A) :- F(~:IA), A being a Borel set in R k. The conditions for convergence ~F,=nk~ {i.e., convergence of the f .d. Fn to the cor responding f .d. stable law #) a re well known (cf. [41]) and we shall not give them. In what follows, condition (A) ((AN)) will denote ,VkF belonging to the domain of (normal) a t t rac t ion of the distr ibution ~kP for all k -> 1. Thus, the basic difficulties a re connected with the proof of the c . c . of the family { Fn, n -> 1}. We recal l that if F is a distr ibution, then by F we denote its cha rac te r i s t i c functional (c. f.). In proving our resul ts the following theorem f rom [1] is important .

THEOREM 13 [1]. Let B be a space of S-type p, l~<p<oo, {v=,aeI} be a family of distributions in B. Let us ~ssume that for each e > 0 there exists a family {? .... aeI} of nonnegative definite quadratic forms of c lass

V. Kapsukas Vilnius State Universi ty . Transla ted f rom Litovsldi Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 20, No. 4, pp. 97-113, October -December , 1980. Original ar t ic le submitted July 25, 1978.

0363-1672/80/2004- 0305507.50 �9 1981 Plenum Publishing Corporat ion 305

p/2, l~<p<2 r~= 1, 2 < p < o ~

, s a t i s f y i n g the cond i t i ons

ao

sup ~ [,%~ (fj)]r~ < 00, a e I .

1 = 1

co

lim sup [q~.,~ (f~)] = 0 . =

and such tha t fo r a l l y~T~ B*

] 1 - R e ; ~ ( y ) [ < ~ , ~ ( y ) + ~ .

Then {v~, ael} is c . c .

Now we s h a l l p r o v e the fo l lowing r e s u l t .

THEOREM 14. Le t B be a s p a c e of S - t y p e p, 1 _ 1 , b e n . i . d . B r . v . wi th d i s t r i b u t i o n F. L e t us a s s u m e tha t t h e r e e x i s t s a s t a b l e d i s t r i b u t i o n p (c~, F) , 0 < a < 2, a nonnega t ive de f in i t e q u a d r a t i c f o r m

q)l on B * , a n d a n u m b e r l _> c~ s u c h tha t one has

I/~ (Y) - ~ (Y) I ~< (71 (y)),/2, y eB*. (2 5)

L e t u s a s s u m e tha t E~ i = 0 i f c~ > 1 and F and F a r e s y m m e t r i c in the c a s e a = 1. L e t %(y)= f (x~ y}~F(dx). Ux

If cond i t i on (AN) holds and q)i, i = 1, 2, a r e of c l a s s r p , then F be longs to the d o m a i n of n o r m a l a t t r a c t i o n o f # .

P r o o f . F i r s t l e t us a s s u m e tha t F and # a r e s y m m e t r i c . S ince Fn(A) = F * n ( n l / a A ) a n d # ( A ) = ~*n(ni/C~A),

one has

I 1

c~

I,-~(y)l-< fl I<x, y>i'r(d~).<c~(~, r)( f <x, y>'r(dx)) ~ (27) U~ Ux

2

We w r i t e ?~,~,(y)_2(C~(~, F) ~-~)~ ?2 (Y), so i t is e a s y to ge t f r o m (27) tha t fo r a l l yeB ~ and e > 0

I I - ~ (Y) t ~< V.~., (Y) + ~.

F r o m (25) we have

_• 1 ~__~ t i ( ) t (28)

2

W r i t i n g %,t(y)=c T~I(Y) and a r g u i n g as a b o v e , f i na l ly we ge t f r o m (26)-~28) tha t fo r a l l yEB* and e > 0

I1 - Re Je.(2) I < ~o,I(Y) + ~., ~(Y) + ~.

W h e n c e , by T h e o r e m 13, fo l lows the c. c. of the f a m i l y { F n , n -> 1~. I t is e a s y to s e e tha t in the g e n e r a l c a s e , ^

i n s t e a d of I1 - ~ ( y ) l i t is n e c e s s a r y to e s t i m a t e I1 - R e / ~ (y)I, w h i c h does not invo lve any d i f f i c u l t i e s , s i n c e

i I f (x, y}[(x, y}[ s 'F(dx) <~ f I(x,Y}[ ~F(dx). The t h e o r e m is p r o v e d . U~ UI

R e m a r k s . 1.. If in (25) l > c~, then th is c o n d i t i o n g u a r a n t e e s the c o n v e r g e n c e of the f . d . (cf . , e . g . , [35] fo r o n e - d i m e n s i o n a l d i s t r i b u t i o n s , and in the g e n e r a l c a s e one c a n app ly the me thod of C r a m e r - W a l d ) . I t is not

c l e a r i f th is i s so in the c a s e l = ~ .

2. I f a -> 1, then the c o n d i t i o n on ~ c o i n c i d e s wi th the c o n d i t i o n g i v e n in T h e o r e m 7 (Sec. 3), but in the c a s e a < 1 th is is a l r e a d y a s u p e r f l u o u s r e s t r i c t i o n , s i n c e a l l s . m. F a r e s . s . m. wi th r e s p e c t to a < 1.

I t s e e m s to us tha t q u a d r a t i c f o r m s a r e w e l l a d a p t e d to l i m i t t h e o r e m s wi th G a u s s i a n l i m i t law~ but a r e not e f f e c t i v e m e a n s in the c a s e of s t a b l e l i m i t l aws . Hence t h e r e i s s e n s e in i n t r o d u c i n g o t h e r f unc t i ons , w h i c h a r e not q u a d r a t i c . L e t ~ be a nonnega t ive h o m o g e n e o u s funct ion of o r d e r l , 0 < l < 2 [ i . e , , r l a l / r on

306

co

B*. We s h a l l s ay that r is of c l a s s p i f ~ (q~ (~,))P< co (we r e c a l l tha t {f j , j _> 1} is a b i o r t h o g o n a l s e q u e n c e in j = l

B*). L e t ~)i,.,~ denote the s e t of nonnega t ive h o m o g e n e o u s func t ions of o r d e r I of c l a s s p, hav ing the fo l lowing

p r o p e r t y :

~, (x + y) ~< ~I' (x) + ,b (y) fo~ x e T,,, y ~ T,,, (2 9)

w h e r e I t and 12 a r e f in i t e s e t s of i n d i c e s such tha t IxNIz= e~ and i f I = { i l , . . . . ik} , then TI deno te the l i n e a r s u b s p a e e g e n e r a t e d by the e l e m e n t s f i l . . . . . f ik. We w r i t e

=1 2p(~p+~(~-;))-~, 1.<;<2, r ( p , l) / 1, p>2.

With the he lp of n o n e o m p l e x but a w k w a r d c a l c u l a t i o n s , wh ich we s h a l l not g ive , one c a n p rove the fo l lowing

ana log of T h e o r e m 13.

T H E O R E M 15. L e t B be of S - type p, 1~<p< 0% {v,, aeI} be a f a m i l y of d i s t r i b u t i o n s on B. L e t us a s s u m e tha t for e a c h e > 0 t h e r e e x i s t s a f a m i l y {4 .... aeI} f r o m ~;C,,,., r = , " (;, [) such tha t

c~

sup ~ (~..~ (fj))~ < co, a E f j = l

co

0 lim sup ~ (*~_:(fj)) = , N~.ao a e l

j = N

[ 1 - Re % (y) I ~< +.,~ (Y) + e

fo r a l l yeT. Then {v,, a~l} is c . c .

B e f o r e app ly ing this t h e o r e m , we f i r s t g ive e x a m p l e s of func t ions ~, c o n n e c t e d wi th s t a b l e l a w s , and we t ry to c o m p a r e T h e o r e m s 13 and 15. L e t

+ (y) = f t<~, y>!~(~x), 0<~<l ,

U~

( f < y , < , <2 U ,

y~B*, (30)

This func t ion is h o m o g e n e o u s of o r d e r c~, and if & _< 1, then fo r a l l y , vcB* ,~(y+v)~ (y)+q~ (v), and if c~ > 1,

then (29) h o l d s , i f f <x,f~> (x , f j> F(dx)=0 fo r i 7 j . But in the c a s e ~ > 1 we have $(y) = (q~(y))C~/2 (cf. T h e o -

1 ] r e m 1 4 ) and s i n c e o:p(2p+~(2-p))- <TP (the c a s e 1 _ 2), i t is e a s y to s e e

tha t the cond i t ions for the func t ion S to be of c l a s s r (p , c~) o r of c l a s s 1 a r e m o r e r e s t r i c t i v e than the c o r r e - r e s p o n d i n g cond i t ions fo r ~ f r o m T h e o r e m 14. But in the c a s e c~ _< 1 we can e a s i l y c i t e e x a m p l e F , fo r w h i c h

wi l l be of c l a s s r (p , a ) , but q~ wi l l not be of c l a s s p / 2 . In f ac t , l e t B=/p, 1 ~<p<2, F (e~)=a. ~ a~< co, and we c a n c h o o s e a i so tha t one has

co oo

1 = l i = t

2p p s i n c e f o r a l l l ~ p < 2 and 0<~<1 , - _gp+~(~ - f>T '

T h e n i t m a k e s s e n s e to f o r m u l a t e the fo l lowing

P r o p o s i t i o n 16. In T h e o r e m 14 in c a s e 1 _< p < 2 and 0 < oe _< 1, # ( a , F) c a n be t a k e n s o tha t the func t ion 4, f r o m (30) w i l l be of c l a s s 2p{2p + c~(2 - p ) ) - l . I n s t e a d of (25), one c a n r e q u i r e tha t fo r a l l yeB*

1 ,e (y) _ ~ (y) I -< L (y),

w h e r e Le!7(,,r, I>r and r = r (p , l).

I t is e v i d e n t f r o m T h e o r e m 14 and P r o p o s i t i o n 16 tha t fo r the d i f f e r e n c e F(y) - ~ { y ) one c a n app ly e s t i - m a t e s of two types . As in the f i n i t e - d i m e n s i o n a l c a s e , the d i f f e r e n c e be tw e e n the c . f . c a n be e s t i m a t e d by m e a n s of p s e u d o m o m e n t s . F o r e a c h ycB* we w r i t e

307

~=(y)=,( I<x, y>l'lF-~l(dx), m>~. /3

This func t ion is def ined if ( [] x ]]'~ I F - ~ I (dx) < ~. It is easy to see that one has B

]['(y)-(zO,)l<<. C~,.(y), :~<m<~l,

I ~ ( y ) - B (y)1 < c~,,, (y), l<o;<<.m<~2.

In the l a t t e r c a se there is n e c e s s a r i l y s a t i s f i ed the n a t u r a l condi t ion: for a l l yeB*

If m _ (r-~)(dx)=O B

j 1 B

If 1 < m < 2, then tim(Y), in g e n e r a l , does not s a t i s fy (29), so we m u s t use/32(y) and r e q u i r e that this quad ra t i c f o r m be of c l a s s rp. Thus , we get the cond i t ion

(.f l<x, (32) j = l B

Finally, we can formulate the following result.

THEOREM 17. Let us assume that #(c~, F) satisfies the hypotheses of Theorem 14 or Proposition 16. Let us a s s u m e

a) in c a s e 0 < a -< 1, /3m(y) ex is t s for some ~ < m-< 1 and s a t i s f i e s (31);

b) in ca se 1 <c~ <2 for al l y E B * , f <x, y ) ( F - ~ ) (dx)=O, ~2(Y) ex is t s and sa t i s f i e s (32). B

Then F belongs to the d o m a i n of n o r m a l a t t r a c t i o n of the s t ab l e taw p(c~, F).

In Banach spaces of S- type p, 1 <_ p < 2, one can apply ano the r approach , based on c r i t e r i a for c o m p a c t - ness of f ami l i e s of in f in i t e ly d iv i s ib l e laws, and developed in [29, 30]. We r e c a l l some concepts f r o m the f i r s t p a r t of the paper . Let M be a p r o p e r Levy m e a s u r e . By cPo is M we denote a p robab i l i ty m e a s u r e on B with c . f . (3). W2 note that if M is a f ini te s y m m e t r i c m e a s u r e , then cPo is M = e(M). F u r t h e r , M / V r denotes the r e s t r i c t i o n of the m e a s u r e M to V,= {x~B : II x [[ <z}, A c denotes the c o m p l e m e n t of the s e t A. In what follows an i m p o r t a n t ro le is played by the fol lowing t h e o r e m , ana logous to T h e o r e m 4.1 of [29].

THEOREM 18. Let B be a Banach space of S- type p, 1 _ 0 and {V~/V~, iel} c,e.;

2) for all yeB*,sup f (y, x>~,(dx)<oo; I e l V~

co

s I { [; )]} 3) lira sup ~1~ 1-exp ( cos ( x , f~}v -1 )~ , (dx d~=O for some q, 0 < q s p. m--+~ l e I

k = m 0 V~

Then ~i is a p r o p e r Levy m e a s u r e for any iEI and the fami ly {cPois i~, is e . e .

Proof . Let us a s s u m e f i r s t that Pi a r e s y m m e t r i c . In view of cond i t ion 1) it suf f ices for us to show that tzt I V1, i~I, a r e p rope r Levy m e a s u r e s and {e (~, I v1), ici} is e . e . Hence to s imp l i fy the no ta t ion we se t ~i(V e) = 0 for a l l ieI. We sha l l show that the f ami ly { e (~, i V~/,) i e I, r/> 1 } s a t i s f i e s the p l a n a r c o n c e n t r a t i o n inequa l i ty , i . e . , for any e > 0 and 6 > 0 the re ex i s t s a f i n i t e - d i m e n s i o n a l s u b s p a c e R ~ B such that

sup sup e(~,/V~lr ) (R,)> 1 - 8 , f E I r~>l

308

w h e r e

We have

R , = { x e B :d(x, R) = inf IIx-zll<e}. z ~ R

oo co

B k - - n + l k = n + I B

(W n is defined in Sec. 3). It is e a s y to v e r i f y that

f } (x , f , ) l 'e(~., /V~l,)(dx)=c f 7-('+~)[1-f cos((x, fk)'r)e(l~,/V1/,)(dx)] d.r<~ B 0 B

co

.<.cf.-'*+"{1-~xp[ f (oo,<x,A>'~-l) v.,(dx)]}a-., (a4) 0 Ilxll ~< l

w h e r e C is s o m e cons tan t . F r o m (33), (34), and condi t ion 3)we get the p lanar c o n c e n t r a t e d n e s s of the fami ly {e(i~i/VT/r), ieI, r> 1}. By L e m m a 1.8 f r o m [29], condi t ion 2 ) a n d C h e b y s h e v ' s i n e q u a l i t y , we get that (e(~dV%) o ./-1,. iel, r> l} is a dense fami ly for any yeB*. By T h e o r e m 2.3 of [1], fo r any ieL the fami ly {e(FJV~,~), r> l} is c. c. and by T h e o r e m 1.6 of [29], Pi is a p r o p e r Levy m e a s u r e fo r any id . Ana logous ly , we get the inequal i ty of p lanar c o n c e n t r a t e d n e s s fo r re (F~), i~I} and the dens i ty of {e (F~}oy -~, i~1} for any yeB* and T h e o r e m 2.3 of [1] g ives the c . c . of the fami ly {e (~,), i~I}. The g e n e r a l c a s e is p roved ana logous ly to the p roof of T h e o r e m 4.1 of [29]. The t h e o r e m is c o m p l e t e l y proved.

We note a c onse que nc e of T h e o r e m 18, which supp lemen t s the th i rd sec t ion . We wr i t e :

A~ = ( t <x, fj:) I p V(dx). Ua

T H E O R E M 19. Le t B b e a B a n a c h s p a c e of S - type p, 1_< p <2 .

I. I f the s. m. F sa t i s f i e s the condi t ion

k ~ l

then F is a s . s . m , wi th r e s p e c t to o~ = p ~ 1.

tI. I f the s . m . F s a t i s f i e s the condi t ion

then F is s . s . m , wi th r e s p e c t to a < p.

m

Z Ak< co, k = l

To prove the t h e o r e m , it is n e c e s s a r y to show that the m e a s u r e M, defined as fol lows:

M(dx)= dr F(ds), s= x r=llxl',

sa t i s f i e s the hypo theses of T h e o r e m 18. The ca lcu la t ions n e c e s s a r y for this a r e analogous to the ca lcu la t ions which w e r e made in the p roof of T h e o r e m 20, so we omi t them. We note a l so that the c a s e o~ > p in spaces of S - type p is c o n s i d e r e d in [40]. Now we give the r e s u l t s on domains of a t t r ac t ion . We wr i t e :

/k,~=sup t~ P{ I (~x, f~}]> t }. t

THEOREM 20. Let B be a Banach space of S - type p, 1 ~ p < 2. The B r . v . ~l belongs to the domain of n o r m a l a t t r a c t i o n of the s t ab le law p with exponent o4 ~e[l, 2) , if condi t ion (AN) holds and a l so the fol lowing condi t ions :

(B) supnP{ll~all>~nX/~'}<m for a l l 5 > 0; n

(C) lim lim sup nP{ [1Vm ~x I[ > tnl/~ } = 0 m--~ao n

309

for some t > O, w h e r e co

Vm x = ~ <x, fk> ek; k = m

(D) a) fo r a p

Qo

i f l ~ < k , e < OD:

k = l

R e m a r k . As shown in [30], if B i s a s p a c e of type p, then c ond i t i ons (AN)-(C) g u a r a n t e e tha t ~1 be longs to the d o m a i n of n o r m a l a t t r a c t i o n of the s t a b l e law wi th exponen t ~ < p. E x a m p l e 6.2 of [17] shows tha t in the s p a c e lp cond i t i ons (AN)-(C) a r e no l o n g e r s u f f i c i e n t i f ~ -> p.

P r o o f . We w r i t e Fn(A) = F ( n l / ~ A ) and we s h a l l show tha t the f a m i l y {,%=nFo. n>~l} s a t i s f i e s the h y p o t h - s e s of T h e o r e m 18. Cond i t ions 1) and 2) of th ls t h e o r e m a r e p r o v e d a n a l o g o u s l y to the p r o o f of T h e o r e m 4.3 of [30], so we p r o v e only cond i t i on 3).

We have

n f ( 1 - c o s ( x , f , . ) ' r n - ~ l ~ ) F ( d x ) < . . n f .r s i n y d y F ( d x ) .<_ l l x l l < n Ue~ J (>:,,r~) ] <d-/<~ 0

0 n

Hence

; { [ ]} 1 V+ ?- 1 - e x p ( (cos(x , f~> , -1 )g . , , (dx) d':~ ~ 1 - e x p - . r ~ Ik ~ f sin y �9 , , - ~ d y d,~ ~<

k = m 0 Vt k = m 0 0

�9 m ) g .

"'~TA--dY l'- k,~ V ~ 1 - e o Y d~: + ~ d': .

k = m 0

I f oe > p, then

(35)

x l - l l ~ k �9 . ab

1 - e d~<~ -~-7~b- 1 - e dr , 0

and f r o m (35) and cond i t i on (D) fo l lows 3) of T h e o r e m 18. L e t a = p. If Ik,c~ >- 1, t hen

' { i / r 1 1 I s in y [ _~ 1 - e o dv ~ -- dy dT. o 0 0

If Ik, a _ 1, then

/ / " ' I s / : i l f 1 s 1 siny 1 Isiny[ dyd~<~Cl[l+llo I -is, - - Ik, ~< l]. 1 - e o - d'= <~ T -~- - dy d'= + ,r 0 0 0 1 0

310

F r o m t h e s e e s t i m a t e s , (35) and cond i t i on (D) fo l lows 3 )o f T h e o r e m 18 for a = p ~ 1. If a < p, then we have

1{ f ]} ~+7 1-e• f ( c o s ( x , f ~ ) ~ - l ) ~ . . ( d x ) dv~ ~ f l (x , f D l ' ~ . ( d x ) < ~ C ~ I~,~. k=m 0 Vt k=m V~ k=m

Thus , f r o m T h e o r e m 18 we g e t the c. e. of the f a m i l y { e P o i s , #n , n -> 1}, and we c o m p l e t e the p r o o f of the t h e o r e m j u s t a s in [30]. The t h e o r e m is p roved .

LEMMA 21. L e t ~ be a s y m m e t r i c B r . v . T h e r e e x i s t c o n s t a n t s C, and C 2 s u c h tha t

(sup, t ~ P {] <~,, f> [ > t })~/~ ~< C, sup E <&'nV ~ - f> ]

for any f from B*, and

Sn (sup t= P { ]l g* l[ > t })*/= <~ C= s u p e r ,v - a - '

w h e r e both i n e q u a l i t i e s make s e n s e only i f the r i g h t s i d e s a r e f in i t e , and S n = {1,1 + ~,2 + �9 �9 �9 + ~l,n, ~l,i, i = 1, 2 . . . . . n, a r e i ndependen t c o p i e s of ~l, a ~ (0, 2).

The p r o o f of L e m m a 21 c a n be c a r r i e d out a n a l o g o u s l y to the p r o o f of T h e o r e m 2.1 of [44].

THEOREM 22. L e t ~, be a s y m m e t r i c Ip r . v . 1 _ be a s t a b l e m e a s u r e wi th exponen t a , a > p. I f (AN) ho lds and a l s o

r

vl (S. A> ~ - S s u p ~ i ~ - ! <c~, n

k = I

then ~ belongs to the normal domain of attraction of the measure #.

5. Speed of Convergence to Stable Laws and the Law of

the Iterated Logarithm

Before proceeding directly to estimates of the speed of convergence to stable laws, we consider one

property of these laws, very closely connected with estimates of the speed of convergence.

Let ~ = (~I, V2 ..... ~n .... ) be an ~ r. v. with stable distribution !, (=, F), 0<~<2, F (x) =P {I[ ~ ]I <x), h(x) = F' (x) (if it exists). Up to now, there were no results concerning smoothness of the distribution of norms

of stable vectors in H, and the theorem formulated below, taken from [20], is the first result in this direction.

In [20] we advanced the conjecture that for all stable measures on H the density h exists on [0, ~) and

sup h (x) < C (~, r ) x

But fo r now we c a n p r o v e only the fo l lowing r e s u l t . We s h a l l s a y tha t F be longs to the c l a s s ~I i f the s . m. F has the fo l lowing p r o p e r t y : t h e r e e x i s t s an N - d i m e n s i o n a l (N _> 2) s u b s p a c e HN such tha t F = P 1 + P 2 and Sr~c_ U 1 N HN, SrzgU~ N H~: (S F is the s u p p o r t of the m e a s u r e F , H ~ is the o r t h o g o n a l c o m p l e m e n t of the s u b s p a e e HN).

THEOREM 23. Let 3 c~I and 1 _< a < 2. Then there exists a constant C(o~, F, N) such that

sup h (x) ~< C (~, F, N). (36) x

The p r o o f of the t h e o r e m is c a r r i e d out a c c o r d i n g to the s c h e m e of p r o o f of L e m m a 1 of [36]. To s i m p l i f y the no ta t ion , Ie t us a s s u m e tha t H N is g e n e r a t e d by the f i r s t b a s i s v e c t o r s {e l , i = 1, 2, . . . . N}. This m e a n s tha t i f ~ eg/, then the v e c t o r s glq = (Vl, �9 �9 � 9 ~N ) and (~N+I, ~N+2, �9 �9 �9 ) a r e i nde pe nde n t . L e t

N 0o

~=~ ~, ~= ~ ~, r~(x)=e{7<x}, t = 1 I = N + I

F= (x) = e { ~ < x }, x >~ O, .f ,(x)=F~ (x).

I t is known (of. [32]) tha t fl (x) ~ x -0+c~) a s x ~ ~o and fl(x) ~ x N-1 as x ~ 0, w h e r e fl i s the d e n s i t y of the d i s t r i b u t i o n ~ . S ince A (x )=(2Vx) -~ f (Vx) , one has tha t f~ (x) i s bounded fo r a l l x -> 0. F u r t h e r , the d e n s i t y

d g (x) =-~x P { Ii ~ Ii ~ < x } c a n be e x p r e s s e d in the fo l lowing way:

311

x

g (x) = ( /'1 ( x - v) dF2 ( y ) (37) (I

whence i t fo l lows tha t g(x) -< C ( e , F , N). S ince h(x) = 2xg(x2), we m u s t show tha t g(x) -< C ( a , F, N)x -1/z fo r

x > C. We use (37):

x x~'2 x co

f. f , (x - y) dF= (y) = .( / , (x - y) dF2 (y) + .( f , (x - y) dF~, (y) < x / 2

0 0 . ~- x ] 2

~<C(cq F, N) x - + P ~ > ~ - .

F r o m [2], w h e r e s t a b l e m e a s u r e s in g e n e r a l v e c t o r s p a c e s a r e c o n s i d e r e d , i t fo l lows tha t fo r any s t a b l e v e c t o r in the B a n a c h s p a c e ~, P{II ~ll > t} < Ct - ~ , so

P ~2> T ~<C(~, P, n) x (39)

F r o m (38) and (39) fo l lows (36) and the t h e o r e m is p roved .

The most general estimate of speed os convergence can be obtained from the following general theorem

on approximations of distributions of two sums in Banach spaces (cf. [37] or [39]). If B is a Banach space and

f: B --RI, then by D(i)f(a) we denote the i-th strong derivative at the point a e B D k will denote the class of all

k times differentiable functions (D O is all continuous functions); D k, k = 0, i, . .., 0 ~< c~ _< I, denotes the class

of those f ~ D k for which

[j D (k) f (x + h) -- D(k) f (x) [i = 0 (!L h [[~) (40)

f o r a t l x e B .

L e t q and ~i, i = 1, 2, . . . , n, b e t w o s e q u e n c e s of i n d e p e n d e n t B r . v. wi th d i s t r i b u t i o n s F i and Gi , i = n n

1, 2, . . . . n, r e s p e c t i v e l y , and F n and Gn be the d i s t r i b u t i o n s of the s u m s ~ ~, and ~ ~,. Le t I = l i = 1

n

, , u) = f !1 x II~l 6 -Gj I(&), ~, (n) = ~ ~, (j). B i = l

THEOREM 24 [37, 39]. Le t the s p a c e B, the B o r e l s e t A c B , a n d t h e d i s t r i b u t i o n s F i and Gi , i = 1, 2, . . . , n, be such tha t for s o m e i n t e g e r m-> 0 and r e a l n u m b e r r , m < r -< m + 1, one has the fo l lowing con -

ditions:

i) for each e > 0 there exist two functions /~.~cD~, y=r-m, 0<~fi,~(x)<~l, i=l,2, with constants in (40),

bounded by Ce -r such that

w h e r e

T h e n

1, x e A , { I, x s A _ ~ , f l ' * (x)= 0, x ~ A ~ , f2'~(x)= 0, x ~ A ,

A ~ = { x e B : ] l x - y l ! < z , y e A } , A_~=((AC)~) ~, A c = B \ A ;

2) for a l l xEB, i=1, . . . , m a n d j = 1 , 2 . . . . . n

f D(Ofg,~ (x) (y)' (F i - G i ) (dy) = O, k = 1, 2, . . . , v, (j) < oo ; B

3) fo r any ~ > O

r I

IF, (A) - G, (A) I ~< C (m) @1 (B, A, ~ , ) )7~ (~, (n));T5 .

R e m a r k . If in (41) i n s t e a d of a one s u b s t i t u t e s a a , O < 6 -< 1, then in (42) i n s t e a d of the exponents r / (r + 1) and 1 / @ + 1) one w i l l have r / ( r + 6) and 5 / ( r + 6), r e s p e c t i v e l y .

(41)

(42)

312

The proof of the theorem is standard and we indicate only the basic steps. Using conditions I) and 3) it

is easy to get the estimate

IF. (A) - G. (A) I ~< max I f f',~ (x) (F. - G.) (dx) l+ Cl (B, A, Gn) ~. (43) /=1, 2 I

B

Further, using the identity expansion we get the estimate

n

I f fi,~ (x)(F.- 5.)(dx)~< ~ I Ws 1, (44) B j= l

w h e r e W s = f hj,~,~ (5') (F~-G~)(dy), the funct ion hj , i , e is d e t e r m i n e d with the help of fi ,a and the d i s t r ibu t ions

Fk . k <- j - 1, and Gi. I ~ j + 1. F u r t h e r , u s ing condi t ion 2) and the T a y l o r expans ion , we get the e s t ima te

I W j I< C(m) ~-' v, (2- (45):

F r o m (43)-(45) we ge t the p roo f of the t h e o r e m .

Here one should note that to estimate the integral

f A,, (x) (Fo - ~.) (a'x) B

one can apply the p r e p a r e d e s t i m a t e s of [45, 46], but fo r this one would have to s t r e n g t h e n condi t ion 2) of the t h e o r e m , s ince it is n e c e s s a r y to include t r ans l a t i ons of the funct ions f i ,e .

F r o m this gene ra l t h e o r e m one can get e s t i m a t e s of the speed of c o n v e r g e n c e to s table laws in Hi lbe r t space .

As the r e f e r e e r e m a r k e d , the e s t i m a t e in T h e o r e m 25 g iven below can a l so be de r ived f r o m T h e o r e m s 2, 3, and 5 of [45].

To simplify the notation, we consider the case of identically distributed summands.

Let P~o~ be a stable distribution in H with exponent (~, 0 < a < 2, }I~ be the class of those #a for which

there exists a constant C(a, F) such that for all ~ > 0

sup~(V,,O<~C(~, F)~, g,,~={x:r<~llxli<r+~} r>~O

If ou r c o n j e c t u r e is c o r r e c t , then 91~ is s i m p l y ai1 s tab le m e a s u r e s on H with exponent o~. But even if the con- j e c tu r e is f a l se , T h e o r e m 23 shows that at l e a s t if 1 <~<2,~1~ is not an empty c l a s s .

Le t ~i, i -> 1, be i . i . d . H r . v. with d i s t r ibu t ion F, E ( 1 = 0, i f i t ex i s t s . Let Q, (A) =P (n -~/~ Z, ~ A}, where

n co

Zn: Z ~l , V r : f [Jx l i r lF--~L~) l (dx) , ml: Z f Xj(F--[s , i ~ l H j = l H

co

i,j=I H

x=(x,, x, ..... x ..... )EH, A.=supJ(Q.-I~.)(V.)I. r>~0

THEOREM 25. Let ~.~9/~ and for some r, c~ <r 2 .

Then �9 ~ 1

A.~<C(~, F)(v.. n - ~ - ) ~ . (46)

P r o o f . It su f f i ces fo r us to show that condi t ion 1) of T h e o r e m 24 holds . Let 1 < ~ < r < 2 (the o ther c a s e s a r e c o n s i d e r e d ana logous ly ) , so we have to show that fo r any t > 0 and e > 0 the re exis ts a funct ion gt,a : H [0, 1] s u c h that

313

1, xeVe, g , , , (x) = 0, x ~ v,+~, II D(1) gr,~ (x) - D (l) g t , , (Y) I[ ~ C e - ~ [] x - y i l ' - l .

Let fe :RI -- [0, i] be defined in the following way: fe(u) = I for u -< 0 and fe(u) = 0 for u >- e, there exist

f~, f ; (O)=Oandl/; (x)-f~ (Y) I< C e - ' [ x - y I v, y = r - 1.

Now we se t

f~,~ (u) =f~ (u - t)andgl. ~ (x) =ft,~ (11 x [1 ), x e H;

with the help of noncompl ica ted ca lcu la t ions one can show that the funct ion gt ,e c o n s t r u c t e d in this way s a t i s - f ies the a b o v e - e n u m e r a t e d r e q u i r e m e n t s . The t h e o r e m is proved.

Now we c o n s i d e r s l ower speeds of c o n v e r g e n c e , when Ur = ~ for al l r > a . We in t roduce the fol lowing c l a s s of functions G6, 0 < 6 -- 1: the funct ion g : R i - -R1 belongs to G6 if

g (x) >/0, g ( - x) = g (x), g (1) = I, g (xl) 4 g (x2), g ( ~ ~< "g (x~) '

O <<. x~ <~ x~.

For example, the function

[ln(l+,x]) ]k In(l+ !-n--~(ll~ D ) ] ~ j , k~ 1, " . . . . . . . . l n 2 - - - - - -

belongs to G6 for any 6 > 0.

' , i I F THEOREM 26. Le t ~ e ~ , m ~ = 0 , i = 0 , [~], ~,,=f llxll~g(IIxll)i _~l(dx)<~, w h e r e g e G ~ , ~>0 , i s a n a r b i t r a r i l y s m a l l number . Then H

l

The proof of this theorem is standard and we omit it. To conclude this section we give a result on the

law of the iterated logarithm for i. i. d. B r. v.

If ~i, i >- I, are i.i.d. H r. v. with symmetric stable distribution g~, then in [34] it was proved that

[IIZ'II~lll~176 X } = 1 , 0 < ~ < 2 , (47) P lim sup ~-~-~-) =e

w h e r e Z, = ~ ~,. In [42] it was r e m a r k e d that (47) holds for any s tab le d i s t r ibu t ion ~t and any Banach i = i

space, since the only thing which is essential in proving (47) is the asymptotic relation ~(V c) ~ r -~ , proved

in [21.

Le t (i, i -> 1 , b e i . i . d . B r . v . , E~ 1 = 0, if it ex i s t s , and ~1 belong to the domain of n o r m a l a t t r a c t i o n of the nondegenera te s tab le law p with exponent e , 0 < a < 2. Le t

/k.=sup [P{n -1/~ [IZ. II<r}-,~(V~)I. r

THEOREM 27. Le t B be any s e p a r a b l e Banach space . If A n _< (g(n)) -1, w h e r e g is a monotone i n c r e a s i n g oo

funct ion such that ~ (g (~r))-i < oo for any 13 > 1, then (47) holds. r = l

The proof of the theorem is carried out by the same scheme as in the one-dimensional ease in [31],

where the law of the iterated logarithm was proved for summands with symmetric stable distribution, so we

shall only indicate the basic steps.

We must show that with probability 1 for any e > 0

n-1/= [I Z, N > (logn) (1+') ~-' a infinite number of times (48)

and

n -l/~ It Z, [I > (log n) (1-') ~-1 Infinitely often. (49),

314

L e t A,,=(iZ.,i>n'~J=Oog.)~+~,~}, n~--[}'], w h e r e / 3 > 1, [x] d e n o t e s the i n t e g r a l p a r t of x, B ~ = { i Z . ii>nj~'(logn.) ~+<'~ fo r s o m e n, n r -< n < n r+ l} .

We have l i m s u p A . c l i m s u p B , . L e t us a s s u m e tha t fo r l a r g e r we have

P (Br) <. CP (D,).

T h e n to p r o v e (48) we m u s t s h o w tha t ~ P(D, )< ~ . One c a n show tha t

(50)

P(D~)< C ( n , ~ - l)nr ~ (iogn,)-(~+ ~'~+g-~ (n,+~- 1),

whence it is easy to deduce that ~ P(D,)< co We write

Y~=Z%~a-Z. ,, E . = { J Y , J]>(n,+a

aa

One c a n show tha t ~ P (E,)= co, so w i t h p r o b a b i l i t y 1,

;[ y~ll>(n~+z_n,)l,=(logn, ) '2 infinitely often.

L e t us a s s u m e tha t (49) i s f a l s e , i . e . ,

P { n J/:' I Z. dl > (log n) I- ~= a finite number of times } > O.

From this and (51) it follows that

(51)

(52)

. . _~ ._ - ' _ - - (~+~ \1/~ , j > ( ~ "~ ;

Further, one can show that for sufficiently large r, the last difference majorizes the quantity (lognr+i) i-C/~ so (49) is proved. To complete the proof of the theorem it remains to prove (50). For this we Use the following assertion, proved in [33] for H r.v., but which can be carried over without change to Banaeh spaces.

LEMMA 28 [33]. Let Xi, i -> l, be independent B r.v. such that for all k < n,

n

j = k

T h e n

i=l i=l

Now in this lemma we set

S h o w i n g t h a t fo r l a r g e r

a - a. = n)/~ (log n,):+ ~.'%

b ~ b. = ((log n,) ~, - 1) (n~ (log nr) 1 + ~,)i/=, ~2 = ~- - ~x..

P{ ' ~ ~ > b r } < y < l i=nr +J

for a l l i _< j < n r+ 1 - n r , f r o m L e m m a 28 we g e t (50). The t h e o r e m is p r o v e d .

315

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infinitely divisible and stable laws in separable banach spaces. ii

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