On a bound of the absolute constant in the Berry--Esseen inequality for i.i.d. Bernoulli random variables

TL;DR

This paper establishes a new upper bound for the absolute constant in the Berry--Esseen inequality for i.i.d. Bernoulli variables, improving previous bounds through computational and analytical methods.

Contribution

It introduces a novel approach combining computational and analytical techniques to bound the Berry--Esseen constant for Bernoulli sums, surpassing prior results.

Findings

The absolute constant is less than the Esseen constant for up to 500,000 summands.

An explicit bound in the local Moivre--Laplace theorem is provided.

The method developed can be applied to other two-point distribution problems.

Abstract

It is shown that the absolute constant in the Berry--Esseen inequality for i.i.d. Bernoulli random variables is strictly less than the Esseen constant, if $1\le n\le 500000$, where $n$ is a number of summands. This result is got both with the help of a supercomputer and an interpolation theorem, which is proved in the paper as well. In addition, applying the method developed by S. Nagaev and V. Chebotarev in 2009--2011, an upper bound is obtained for the absolute constant in the Berry--Esseen inequality in the case under consideration, which differs from the Esseen constant by no more than 0.06%. As an auxiliary result, we prove a bound in the local Moivre--Laplace theorem which has a simple and explicit form. Despite the best possible result, obtained by J. Schulz in 2016, we propose our approach to the problem of finding the absolute constant in the Berry--Esseen inequality for two-point distributions since this approach, combining analytical methods and the use of computers, could be useful in solving other mathematical problems.