On the characteristic function of a sum of m-dependent random variables (Q1088277)

Let \(S=X_ 1+X_ 2+...+X_ n\) be a sum of 1-dependent random variables with zero mean and let \(\sigma^ 2=ES^ 2\), \(L=\sigma^{- 3}\sum^{n}_{k=1}E| X_ k|^ 3\). It is proved that there is a universal constant a such that for \(a| t| L<1\) the following inequality holds \[ | E \exp (it\sigma^{-1}S)| \leq (1+a| t|)\sup \{(a| t| L)^{-(1/4)\ln L},\quad \exp (-t^ 2/80)\}. \] The Shergin method of decomposition of S is used. With the help of this estimate a Berry-Esseen theorem is proved.