I. G. Shevtsova

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

By a modification of the method that was applied in study of Korolev & Shevtsova (2009), here the inequalities and are proved for the uniform distance ρ(F n ,Φ) between the standard normal distribution function Φ and the distribution function F n of the normalized sum of an arbitrary number n≥1 of independent identically distributed random variables with zero mean, unit variance, and finite third absolute moment β3. The first of these two inequalities is a structural improvement of the classical Berry–Esseen inequality and as well sharpens the best known upper estimate of the absolute constant in the classical Berry–Esseen inequality since 0.33477(β3+0.429)≤0.33477(1+0.429)β3<0.4784β3 by virtue of the condition β3≥1. The latter 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 0.3041 which is strictly less than the least possible value 0.4097… of the absolute constant in the classical Berry–Esseen inequality. As corollaries, the estimates of the rate of convergence in limit theorems for compound mixed Poisson distributions are refined.

On the Upper Bound for the Absolute Constant in the Berry–Esseen Inequality

This paper describes the history of the search for unconditional and conditional upper bounds of the absolute constant in the Berry–Esseen inequality for sums of independent identically distributed random variables. Computational procedures are described. New estimates are presented from which it follows that the absolute constant in the classical Berry–Esseen inequality does not exceed 0.5129.

Sharpening of the Upper Bound of the Absolute Constant in the Berry–Esseen Inequality

The upper bound of the absolute constant in the classical Berry–Esseen inequality for sums of independent identically distributed random variables with finite third moments is lowered to

On the Asymptotically Exact Constants in the Berry–Esseen–Katz Inequality

C\leqslant 0.7056

On the absolute constants in the Berry-Esseen-type inequalities

.

On the Accuracy of the Normal Approximation to Compound Poisson Distributions

A detailed classification of the asymptotic constants in the Berry–Esseen–Katz inequality for the sums of independent identically distributed random variables with finite absolute moments of order

On Nonuniform Convergence Rate Estimates in the Central Limit Theorem

2+\delta

A new moment-type estimate of convergence rate in the Lyapunov theorem

with

On the accuracy of the approximation of the complex exponent by the first terms of its Taylor expansion with applications

\delta\in[0,1]

On the Absolute Constants in Nagaev–Bikelis-Type Inequalities

is proposed. Two-sided bounds and/or exact values for the asymptotically best and asymptotically exact constants are given. The lower bounds for the exact constants for the case