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 rho(F-n, Phi) <= 0.33477(beta(3) + 0.429)/root n and rho(F-n, Phi) <= 0.3041(beta(3) + 1)/root 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 >= 1 of independent identically distributed random variables with zero mean, unit variance, and finite third absolute moment beta(3). 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(beta(3) + 0.429) <= 00.33477(1 + 0.429)beta(3) <0.4784 beta(3) by virtue of the condition beta(3) >= E1. 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.