An improvement of the Berry--Esseen inequality with applications to Poisson and mixed Poisson random sums

TL;DR

This paper improves the Berry--Esseen inequality bounds for sums of i.i.d. random variables and applies these results to refine convergence rate estimates for Poisson and mixed Poisson random sums.

Contribution

It introduces sharper inequalities for the Berry--Esseen bound and applies them to better estimate convergence rates in Poisson-related limit theorems.

Findings

New inequalities with constants 0.335789 and 0.3051 are established.

The improved bounds are sharper than previous best known estimates.

Applications include refined convergence rate estimates for Poisson and mixed Poisson sums.

Abstract

By a modification of the method that was applied in (Korolev and Shevtsova, 2009), here the inequalities $$\rho(F_n,\Phi)\le\frac{0.335789(\beta^3+0.425)}{\sqrt{n}}$$ and $$\rho(F_n,\Phi)\le \frac{0.3051(\beta^3+1)}{\sqrt{n}} $$ are proved for the uniform distance $\rho(F_n,\Phi)$ between the standard normal distribution function $\Phi$ and the distribution function $F_n$ of the normalized sum of an arbitrary number $n\ge1$ of independent identically distributed random variables with zero mean, unit variance and finite third absolute moment $\beta^3$. The first of these inequalities sharpens the best known version of the classical Berry--Esseen inequality since $0.335789(\beta^3+0.425)\le0.335789(1+0.425)\beta^3<0.4785\beta^3$ by virtue of the condition $\beta^3\ge1$, and 0.4785 is the best known upper estimate of the absolute constant in the classical Berry--Esseen inequality. The second inequality is applied to lowering the upper estimate of the absolute constant in the analog of the Berry--Esseen inequality for Poisson random sums to 0.3051 which is strictly less than the least possible value of the absolute constant in the classical Berry--Esseen inequality. As a corollary, the estimates of the rate of convergence in limit theorems for compound mixed Poisson distributions are refined.