A rate of convergence in the poissonian representation ... - Springer Link

Lithuanian Mathematical Journal, V~d.36, No. 4, 1996

A RATE OF CONVERGENCE IN THE POISSONIAN REPRESENTATION OF STABLE DISTRIBUTIONS M. Ledoux and V. Paulauskas 1. I N T R O D U C T I O N AND S T A T E M E N T O F T H E RESULTS The classical Berry-Esseen theorem gives a rate of convergence in the central limit theorem (CLT) which, in its simplest form, can be formulated as follows (nowadays the name of Berry-Esseen refers to a family of much more general estimates). Let X, Xl . . . . . X,, . . . . be independent identically distributed (i.i.d.) real random variables with mean zero and variance one. Let qb be the standard normal distribution function and S,, = (1/,,/-if) ~ i ~ l Xi. Then the CLT means that the distribution of S,,, £(Sn), converges weakly to qb and the Berry-Esseen theorem asserts that there exists a universal constant C > 0 such that, if/~3 = E([XI 3) < ec, then, for every n, A,, = sup ]P{S,, < x} - qb(x)[ ~< Cfl3/,,/-~

(1)

x

(see [I, 3 or 15]). Later on, this result was generalized in many directions (we refer the reader to the books [15, 17]). For our further purposes, we mention here two generalizations. The first one is a rather easy generalization. If, instead of r3 < e~z, we assume that/32+8 = E(IX[ 2+a) < (:x3 for some 0 < ~ ~< 1, then, for every n, An ~ Cflz+6/rt ~/2

(see [15] for details). The second one is more essential and much more difficult. Let us consider summands Xi having infinite variance. Then we come to a limit theorem with a stable limit law ~a, c~ being the exponent of the law, 0 < c~ < 2. This limit theorem in the simplest case states that, under appropriate conditions (the tails of the distribution of X should be "similar" to those of G~), the distribution of normalized sums (1/n l/~) ~-~in=lXi converges weakly to ~ . Through efforts of many people, rates of convergence in this "stable" limit theorem are now available (we refer for example to the books [2] and [18], where one can find the most important results in this direction). But the nature of rates of convergence to stable laws is quite different from its Gaussian counterpart. Moment conditions are no longer useful and their role is played by pseudomoments. Strict formulations will be given below (see Theorem 6). Let us simply mention at this point that the closer the tails of X to the tails of G~, the better the rates. On the other hand, many results of the literature of the past years have been devoted to the study of stable laws via their Poissonian representation. This representation (which we briefly recall below) goes back to R Ldvy and had been revived in the 1980s by R. LePage (see [9-11]). This representation is true in a general Banach space setting and is very useful in many problems connected with stable laws (see, for example, monographs [7, 8, 13, 16]). Let us first recall this representation. For simplicity, we deal with symmetric stable distributions only. Thus, for 0 < ~ < 2, let ~c, stand for the c~-stable symmetric distribution with Fourier transform ~'~,(t) = D6partement de Math6matiques, Universitd Paul-Sabatier, 31062 Toulouse, France. Vilnius University, Naugarduko 24, 2006 Vilnius, and Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 36, No. 4, pp. 486-500, October-December, 1996. Original article submitted February 7, 1996. 0363-1672/96/3604-0388515.00

388

© 1997 Plenum Publishing Corporation

A rate of convergence in the Poissonian representation

389

exp(-ltl'~/2), t E R. It is known (see, for example, [8]) that lira x'~(1 - G=(x)) = C,~, where C~, > 0 only depends on u. Let ?'l, Y2. . . . be independent random variables with common exponential distribution P{Yl > x} = e -x, x >10. For every j >/ 1, 'set l"j = )-1 + " " + )-j. The sequence (Fj) corresponds to the successive times of jumps of a standard Poisson process. Let X, Xl, X2 . . . . be another sequence of i.i.d. real symmetric random variables with a distribution function F and finite or-moment E(IX]'~). (Here and below the symmetry is assumed mainly to make presentation clearer and to avoid technical difficulties. This Poissonian representation is, indeed, also valid in the general case (see [11, 16]).) Assume that these random variables are normalized so that E(IXI °) = 2C,~ and that the sequences (l-"j) and (X j) are independent. Then the representation theorem says that the series oo

Z

r-fi/aXj

(2)

j=l n

converges almost surely and has distribution Go. Hence the distribution of ~ j = l F f i / ~ X J converges weakly to ~,~. A natural question then is to estimate the rate of this convergence, namely, to estimate the quantity

sup P

~/~xj

< x

-

-

A,(,~, F).

(3)

j=l The main observation of this note is that the rate of convergence in this weak convergence resembles very much Berry-Esseen's rate in the classical Gaussian CLT. Namely, the preceding Poissonian representation only requires otth moment of the summands and we can therefore expect a result very similar to the Gaussian CLT. In order to estimate quantity (3), we assume as in (1) some stronger moment condition on X. To state our results, we use the following notation. Let f r ( M ) , r > cz, denote the class of symmetric distributions on the real line with a finite rth moment not exceeding M and such that E(IXI =) = 2C,~. Let furthermore -~r.c(M, y, fl), 0 < y < 1, fl > O, denote the subclass of.Yr(M) such that the characteristic function F(t) of F E .Tr,c(M, y, fl) satisfies the Cramer condition

I?(t)l
ft.

(4)

One of our main results is the following. THEOREM 1. a) Let ¢z < r ~< 2. For each distribution F E .~r.c(M, Y, fl), there exist constants Ct > 0 and K > 0 depending on M , y, fl, et such that, for all n >f 1, A.(u, F) 0 and r < 2, there exist a distribution F E f r . c ( M , y, fl) and a constant C2 > 0 depending on the distribution F and e such that

A,,(~, ~ )

>1

C~.n-~ -~.

(5)

c) For r = 2 we can omit e in (5). For the particular case of the normal distribution • (with r = 2) we have the estimates from above and from below of the same order. PROPOSITION 2.

There exist positive constants C3 and C 4 depending on c~ only such that 2-or

C3n

~

2 -ct

~ A n ( ~ , (I)) ~ C 4 n

The estimate from above is the main difficulty in the proof of Theorem I. This estimate is proved by the classical Fourier method (see Proposition 10). Without Cramer's condition (4), we are only able at this point to

390

M. Ledoux and V. Paulauskas

prove slower rates of convergence. Namely, by a different approach (avoiding characteristic functions and using the fact that in distribution (r'n+l/n) l/~ Y'~.]=t Ffl/~XJ is equal to (1/n l/~) Y'~I'=I ~iXi for some well-chosen i.i.d. ~i, i /> 1, independent of the Xi's), we can prove the following result. THEOREM 3. Suppose that X is a symmetric random variable with a distribution function F, 1 < ot < 2, E(IXI ~) = 2Cc, and E(IX[ r) < oo for some el < t 0 A,(~, F) = O ( max (\ -l °- -g~n , n

r-~4-~)) .

(6)

From this result we see that, taking r = 2, A,, (c~, F) = O (n-×(~)+~ (log n )/~(cn)

(7)

with 1/2, 2-c~/o~,

Y(~)=

1 < ~ ~< 4/3, 4/3 < ~ < ~ 2 ,

and 1 0

/3(~) =

if if

?,(or)= 1/2, y(~) < 1/2.

From the proof it will be seen that we can consider the case 0 < ot < 1 as well, with however stronger conditions on X. For the previous work on the rate of convergence (here it may be appropriate to mention that the above formulated results were obtained in 1989-90, see acknowledgment at the end of the paper) we refer to [5-7]. In [5] and [6], using the simple relation eo

_

2

o 0 such that, on the set A,

dO

-- dO

/> Cl(O "2 -O'n).

(15)


393

From (14) and (15) we get that, for x > 0,

.,((:) (I) X

> ci f(~2

_ ~r~)dPl

A

/S

Se-*l S (''-'~)dPl}

/> Ci "~I (0.2 -- ¢r~)dPt --{m,

I

(16)

/> CI/El(O "2 - 0.2 ) - - ( E , ( o "2 i~ff)2)l/2[(P,{~ n < o'i) 1/2--~ (Pl{Cr 2> 6-1})1/2]}

[

(where in the last inequality we have used the Cauchy-Schwarz inequality). Now, for every /3 > O, Stirling's formula implies that E1(I-'~) "- j - 3 as j --+ oz. Therefore, there exist constants 0 < C2 ~< C3 < oo such that, for all j >/jo,

c,j-~/,, ')

j.l=n+l

(e,(U4/=)):(e,r,-4/~) 1/2

(18)

j,l=n+l E~} ,

~ "1{0.2 < ~E21~- V~3(2@ff)l/2n ½(l-2'°t,. Bound (20) and hence the conclusion of the proposition easily follow from (19). The proof is complete. [] Remark. The upper bound for the left-hand side of inequality (13) follows immediately using similar estimates. Namely, we can write, for every x > 0,

E t ( q ) ( ~ n ) _ ~(x )) 0. Collecting all these estimates we get

{~.~>~}

f (x-x,,)aPt}. {x>~-t}

(21)

395


Now in the same way we got (17) and (18). we have

2_, e ( r f f ' )

El (X -- ),,,) =

.< csa

j=n+ I

-~, (~ + 1 ) - - (/-

(22)

and

03 O0 (~___~) El ((), - ~.n) 2) E-I))I/2] }.

The rest of the argument is exactly the same as in Proposition 8 and therefore we omit the details. We turn to the upper bounds. PROPOSITION 10. Let Z, Zi, i ) 1, be i.i.d, random variables with a distribution F, E ( Z ) = 0, E(Z) 2 < oo and satisfying Cramer's condition (4) with parameters g < 1 and 15 > O. Then there exist constants C > 0 and v > 0 depending on a and F such that, f o r every n ) no, (24)

An(or, F)