Asymptotically exact constants in natural convergence rate estimates in the Lindeberg theorem

TL;DR

This paper classifies and analyzes the asymptotically exact constants in convergence rate estimates of the Lindeberg CLT, providing bounds and structural improvements for inequalities involving moments and the Lindeberg fraction.

Contribution

It introduces a detailed classification of asymptotically exact constants in Lindeberg CLT estimates and offers bounds and structural improvements for related inequalities.

Findings

Lower bounds for asymptotically exact constants are established.

Universal and optimistic constants are close to upper bounds.

Structural improvements to classical inequalities are presented.

Abstract

Following (Shevtsova, 2010) we introduce detailed classification of the asymptotically exact constants in natural estimates of the rate of convergence in the Lindeberg central limit theorem, namely in Esseen's, Rozovskii's, and Wang-Ahmad's inequalities and their structural improvements obtained in our previous works. The above inequalities involve algebraic truncated third-order moments and the classical Lindeberg fraction and assume finiteness only the second-order moments of random summands. We present lower bounds for the introduced asymptotically exact constants as well as for the universal and for the most optimistic constants which turn to be not far from the upper ones.