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

TL;DR

This paper improves bounds on the constants in Berry-Esseen inequalities for sums of i.i.d. random variables, sharpening the estimates for the absolute constant and providing new inequalities with better constants.

Contribution

The authors introduce modified inequalities that tighten the bounds on the absolute constant in the Berry-Esseen inequality for i.i.d. variables.

Findings

Improved upper bounds for the absolute constant in Berry-Esseen inequality to less than 0.4756 and 0.4748.

New inequalities that refine the classical Berry-Esseen bounds.

Enhanced structural understanding of the constants involved in normal approximation.

Abstract

By a modification of the method that was applied in (Korolev and Shevtsova, 2010), here the inequalities $\Delta_n\leq0.3328(\beta_3+0.429)/\sqrt{n}$ and $\Delta_n\leq0.33554(\beta_3+0.415)/\sqrt{n}$ 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 $n\geq1$ 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 $0.3328(\beta_3+0.429)\leq0.3328\cdot1.429\beta_3<0.4756\beta_3$ by virtue of the condition $\beta_3\geq1$. 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 $C_0<0.4748$.