AN EDGEWORTH EXPANSION FOR A SUM OF M-DEPENDENT RANDOM VARIABLES

Given a sequence XI, X 2 X of m-dependent random variables with n moments of order 3+ (0<<I), we give an Edgeworth expansion of the distribution of -I 2 2 So (S-XI+ X2+...+ X ES under the assumption that E[exp(itSo )] is small n’ away from the origin. The result is of the best possible order.


I. INTRODUCTION.
A sequence of random variables (r.v.) XI, X 2 X is said to be m-dependent n if for each i.<j.<n-m-1 the two sequences (Xi)i.<jand (Xi)i>j+m are independent.Let 2 XI+...+X ES 2 A Berry-Esseen bound of the exact order for the distribution S n' -I of So has been obtained by V.V. Shergin [I] under the assumption of existence of moments of order 2+ (0<<I).
The purpose of this work is to establish an Edgeworth expansion for the -I distribution of So under the assumption of the existence of moments of order 3+ -I (0<<I).Provided that the characteristic function of Eexp(itSo is small away from the origin, this bound is of the best possible order (O(n (I+)/2) in the ,tationary case).The result is stated and discussed in section 2, section 3 outlines the proof nd section 4 contains-+he various estimates needed.Let (t) and (t) be the distribution and density function respectively of a standard normal random variable.Let E(S3)0 -3, and then L (m+1)2 3 3, 2+o-3-e BYrd, where B is some constant, so we get M of the order n -(I+)/2 L of the order -I/2 -5/2 n and N of the order n So, in A, the main term is M (or the main terms are 2 L and M if I).So, provided 6 is small enough the bound given by theorem A is -(I+)/2 of order n the best possible order.
Theorem A gives a first order expansion, but it is clear that the same type of methods will apply for higher order expansion, though the computation becomes rather complex.

METHODS.
We first suppose m=1.We will use the following estimate, proved by the author in [2], using ideas of V.V. Shergln: LEMMA I.There exists a universal constant K2, such that if S is a sum of m- dependent random variables then for K21tlL < I, we have Sup IP(Sot) (t) /6 (1-t2 To study J(t) for Itl =< T O let f(t) E exp(itS0-1).j<nT Y,o 0<_-<511 (exp(it(Uj,-Uj,+1)-1)exp(itUj,6) Except for the last term, the last exponential in each term is independent of the first part of the term.The first term has expectation zero.For the second and third, we expand the expectation of the first part, then replace E(exp(itUj,k)), (k=2,3), by E(exp(itU)), modulo a perturbation.We use several time the estimate (3.1) for these computations.The last two terms are bounded more directly.The result is a relation of the type f'(t) where R(t) and H(t) are small.Integration of this relation yield the needed estimate for f(t).
The method just described has been used in the stationary case by A.N.
Tikhomirov [3].It does not seem possible to extend his method directly to the general case.However, the estimate (3.1) made this possible.It should be noted that the method used to obtain (3.1) does not seem to extend to establish theorem A.