A Non-Uniform Bound of the Remainder Term in the Central Limit Theorem for Bernoulli Random Variables

Abstract

A bound for the remainder in the Esseen expansion is obtained in the case of Bernoulli random variables. The bound consists of two parts, uniform and non-uniform. The uniform part depends only on n and p, and the non-uniform part depends also on x. This bound is compared with other known bounds. It is shown how this result can be applied to the problem of the absolute constant in the Berry–Esseen inequality.