ON THE CHARACTERISTIC FUNCTION OF A SUM OF M-DEPENDENT RANDOM VARIABLES WANSOO

Let S = f 1 + f 2 + … + f n be a sum of 1 -dependent random variables of zero mean. Let σ 2 = E S 2 , L = σ − 3 ∑ 1 ≦ i ≦ n E | f i | 3 . There is a universal constant a such that for a | t | L 1 , we have | E exp ( i t S σ − 1 ) | ≦ ( 1 + a | t | ) sup { ( a | t | L ) − 1 / 4 ln L , exp ( − t 2 / 80 ) } . This bound is a very useful tool in proving Berry-Esseen theorems.

WANSO0 T. RHEE -I variables.We will estimate the bound of IE[exp(itSe )]I by Shergin's methods.This result extracts the most important ideas of Shergin's work.Also we want to point out that this estimate turns out to be an essential tool in the proof of Berry-Esseen type bounds in other limit theorems for m-dependent random variables.In a subsequent work, we shall establish such a convergence rate for U-statistics [2] and an Edgeworth expansion for a sum of m-dependent random varlables [3].

CONSTRUCTION.
We follow the lines of Shergin's ingenious construction to decompose S in an amenable way.We do not however assume the reader to be familiar with Shergin's paper.The exposition is self contained, and some long details of h p Jof at._ eliminated by our approach.
We assume now on m=1.We denote ao,aI,...,a universal constants.No attempt is made at finding optimal values for these universal constants, .ince the numerical values involved here are too large to be of any interest.
The construction stops at an index h such that either s(h) n or
PROOF.From the l-dependence of the f it follows that oi h-1 2 2+ we have p <-(h-I)/I0.It follows that -I there are at least 9(h-I)/I0 indices for which i .>IO, we have H.>R/IO.This follows from the fact that h=>0R/11 and straightforward computations.We can moreover select indices il,...,i H such that for 1<<=h, For < < H we set s(i) s(i+1) a:.We have Let --(a a) -I .
Here the maximum is taken over all possible choices of b s.
It is well worthwhile to reformulate the above result to show more precisely the behaviour of the bound.

REMARK. (I)
In case of a m-dependent (m>1)sequence of random variables, an -I estimate of IE(exp itS )I can be obtained by considering S as the sum of l- dependent blocks of f.. (2) The constant I/4 in the exponent of (4.2) plays no particular role.It is clear from the method that it can be replaced by any number; but the values of a 5 and a 8 depend on this exponent.However for the applications we have in mind, any positive number will be sufficient.
To support our claim that theorem 6 is useful tool, we deduce Shergin's theorem in a simpler way.Let be the distribution function with the normal law.