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

Abstract

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.

Keywords:

• Central limit theorem
• Berry–Esseen inequality
• Smoothing inequality
• Poisson random sum
• Mixed Poisson distribution

MSC classes:

• 60F05

Acknowledgements

The authors have the pleasure to express their gratitude to Professor Jan Grandell whose useful remarks undoubtedly improved the presentation of the material, to Professor Allan Gut whose bibliographical support cannot be overestimated and to Margarita Gaponova who carried out the supplementary computations resulting in Lemma 7. This research supported by the Russian foundation for Basic Research, projects 08-01-00563, 08-01-00567, 08-07-00152 and 09-07-12032-ofi-m, and also by the Agency for Education of Russian Federation, state contracts P-1181 and P-958.