Rate of convergence in the multidimensional limit ... - Springer Link

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on the estimates of the rate of convergence in limit theorems for the sum of independent ... a few articles by I. Banis (see [2-5]) where the rate of convergence isĀ ...
RATE

OF CONVERGENCE

LIMIT

THEOREM

WITH

IN T H E A STABLE

MULTIDIMENSIONAL LIMIT

LAW

V. P a u l a u s k a s

w

Introduction

UDC 519.21

and

Formulation

of Results

AS o b s e r v e d in [17], t h e r e have been published r e c e n t l y and a r e still being published many a r t i c l e s on the e s t i m a t e s of the r a t e of c o n v e r g e n c e in limit t h e o r e m s for the sum of independent r a n d o m variables in the c a s e of the n o r m a l limit law. In the p r e s e n t a r t i c l e we continue the r e a l i z a t i o n of the problem posed in [17], namely, d e c r e a s i n g the gap in our knowledge in the domain of r e m a i n d e r t e r m s in the c a s e of the n o r m a l law and other stable laws, We o b s e r v e with satisfaction that in the one-dimensional case, by virtue of the a r t i c l e s of V. M. Z o l o t a r e v [9], K. I. Satybaldina [21], N. Kalinauskaite [12], S. Staishunas [22], the author [16-18], and a s e r i e s of others, this gap has somewhat d e c r e a s e d since a large number of r e s u l t s about the r e m a i n d e r t e r m in the central limit t h e o r e m a r e obtained as p a r t i c u l a r c a s e s of general r e s u l t s on the r a t e of c o n v e r g e n c e to the stable law. Relatively little consideration has been given to the r a t e of convergence to multidimensional stable laws on account of sufficiently complicated s t r u c t u r e of the latter and s e v e r a l other difficulties. T h e r e a r e a few a r t i c l e s by I. Banis (see [2-5]) where the r a t e of convergence is considered in the case of identically distributed summands and where a l m o s t all methods applicable for the estimation of r e m a i n d e r t e r m s in the multidimensional c a s e a r e tried. Unfortunately, all the e s t i m a t e s have deficiencies and do not differ by the generality of the formulation of the problem. As a rule, in the e s t i m a t e s , only the dependence of the r e mainder t e r m on the number of t e r m s n is c o r r e c t . In the p r e s e n t a r t i c l e the p r o b l e m of the estimation of the r a t e of convergence to multidimensional stable laws is posed in the following form. Let F n be a distribution of the n o r m a l i z e d sum of identically distributed independent r a n d o m v e c t o r s with a distribution F, let Got be a stable k-dimensional distribution, and let G be a c l a s s of bounded m e a s u r a b l e functions. Under c e r t a i n assumptions about the distribution F we seek an e s t i m a t e of the type sup g~G

g (x) (F, - G~) (dx) ~O, Im '~=i~+ ... +ik. L e t u s put

k

169

a~(G~, l)=

sup

f '~D,.g~(x, 1):dx

sup

m =I .....

1 il+...+ik~ 0

~

[m l

Rk

and let a 3 (Go/, k) be a c o n s t a n t f o r which

G~(S~, I, ~j, ~, l)~ 0 . T h e e x i s t e n c e and p r o p e r t i e s of a 2 (Go/,

L e t F, (A)=P

Z

(2)

) and a 3 (Go/, k) will be c o n s i d e r e d in w

~ ~-A,,)EA , w h e r e A n = 0 if a ~ 1 and A n is defined by ( 1 3 ) f o r ~ = 1. i=l

Now we c a n f o r m u l a t e o u r f u n d a m e n t a l r e s u l t excluding the n o r m a l d i s t r i b u t i o n f o r the t i m e being f r o m o u r c o n s i d e r a t i o n , i.e., a s s u m i n g that 0 < ~ < 2. T H E O R E M 1. L e t the following conditions be fulfilled for an i n t e g e r m, [a] - m . l .

Rk

L e t us deduce s o m e of the m o s t i m p o r t a n t c o r o l l a r i e s f r o m the t h e o r e m f o r m u l a t e d above. L e t G 1 (d) denote the ' c l a s s of functions s a t i s f y i n g the conditions ] g (x) [ 0; t h e r e f o r e , in w we indicated all t h e s e p a r a m e t e r s e x c e p t the v a r i a b l e t o r the s e t A within the b r a c k e t s in the e x p r e s s i o n of the c h a r a c t e r i s t i c function or the d i s t r i b u t i o n of the s t a b l e law, but in c a s e this will not c a u s e m i s u n d e r s t a n d i n g we will omit all o r s o m e of t h e s e p a r a m e t e r s . It is obvious f r o m (9) that l f l a ( t ) I ~8-~-a~(G, G~)/,=-~1 8 - ~ > 0 .

- I Rk

Then

8">1

g~',,(x)H*G~,(dx, ~,) = Rk

I: f g:''~(''+z) H(dx)G~ {dz, )') ~ ~ .: ( (g~'~(X+z)H(dx)) G'( dz " ~ ~-l;z .~h

Rk

Rk

S i n c e we c a n l e t 8 t e n d to z e r o , we h a v e 8">~-I 8 - - ~4 ~G~, (Sh, D.

(15)

L e t u s c o n s i d e r t h e f o l l o w i n g two c a s e s s e p a r a t e l y . I

l

, on a p p l y i n g the e s t i m a t e (2) we g e t f r o m hZ

(15) 8 4 a~ (k, G~) l 8"~>~ 8.

1

C a s e II.

I

,k Go))-'.

In t h i s c a s e w e s i m p l y e s t i m a t e Got (S~, },) -< 1 a n d m a k e u s e o f t h e

c h o i c e of h: 1

8" >I $ - 3- 9 ~- al (G, G~) h >I -~ - 2ai

(G, G~)

I

(G~,

177

H e n c e (14) f o l l o w s , T h e l e m m a i s p r o v e d . Remark.

If a > 1, t h e n

G~(SL i)~ 1, t h e n we c a n u s e t h e f o l l o w i n g e s t i m a t e in p l a c e o f (21): m

k2 . < 2 -,,:,! - -,',.sup '.r,,,~(v) ,i=,, ',Dtki(x):[lyil ~.

In w h a t f o l l o w s w e w i l l a l s o a s s u m e t h a t )t < 1; t h e n ( n . i - 1 ) / n

+ ~t < 2, a n d w e g e t

!

rmi(y)lh

n~

sup

--~

t 9