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 β
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(β
3+0.429)≤0.33477(1+0.429)β
3<0.4784β
3 by virtue of the condition β
3≥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.
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.