On the absolute constants in the Berry-Esseen type inequalities for identically distributed summands

Abstract

By a modification of the method that was applied in (Korolev and Shevtsova, 2010), here the inequalities ${\Delta }_n\le 0.3328({\beta }_3+0.429)/\sqrt{n}$ and ${\Delta }_n\le 0.33554({\beta }_3+0.415)/\sqrt{n}$ are proved for the uniform distance ${\Delta }_n$ between the standard normal distribution function and the distribution function of the normalized sum of an arbitrary number $n\ge 1$ of independent identically distributed random variables with zero mean, unit variance and finite third absolute moment ${\beta }_3$. The first of these two inequalities improves one that was proved in (Korolev and Shevtsova, 2010), and as well sharpens the best known upper estimate for the absolute constant $C_0$ in the classical Berry--Esseen inequality to be $C_0<0.4756$, since $0.3328({\beta }_3+0.429)\le 0.3328\cdot 1.429{\beta }_3<0.4756{\beta }_3$ by virtue of the condition ${\beta }_3\ge 1$. The second of these inequalities is also a structural improvement of the classical Berry--Esseen inequality, and as well sharpens the upper estimate for $C_0$ still more to be $C_0<0.4748$.

7 pages

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