On Convergence Rate in CLT for Smooth Distributions

Abstract

There are obtained nonuniform estimates of the first and second order for the rate of convergence in the central limit theorem for sums of independent identically distributed random variables with bounded density and finite absolute moments of order 2+δ, where 0 < δ ≤ 1. The obtained estimates represent a sum of two terms, the first one being the Lyapunov fraction of order 2 + δ with a factor depending only on δ, and the second one decaying exponentially. The values of the factor of the Lyapunov fraction are considerably less than the known. This is an updated version of the original paper.