On the characteristic function of a sum of M-dependent random variables

Abstract

Let S=f1+f2+…+fn be a sum of 1-dependent random variables of zero mean. Let σ2=ES2, L=σ−3∑1≦i≦nE|fi|3. There is a universal constant a such that for a|t|L<1, we have|Eexp(itSσ−1)|≦(1+a|t|)sup{(a|t|L)−1/4lnL,   exp(−t2/80)}.This bound is a very useful tool in proving Berry-Esseen theorems.