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

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≤n≤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.

Keywords

• Optimal value of absolute constant in Berry--Esseen inequality
• binomial distribution
• numerical methods