Esseen-Rozovskii type estimates for the rate of convergence in the Lindeberg theorem

TL;DR

This paper improves classical estimates for the rate of convergence in the Lindeberg theorem, providing explicit constants, analyzing key fractions, and establishing new inequalities for distribution tails.

Contribution

It offers structural enhancements of Esseen's and Rozovskii's estimates, computes explicit absolute constants, and introduces asymptotically exact constants in convergence bounds.

Findings

Derived upper bounds for convergence rate constants.

Compared Esseen's, Rozovskii's, and Lyapunov's fractions.

Proved a sharp inequality for quadratic tails of distributions.

Abstract

We present structural improvements of Esseen's (1969) and Rozovskii's (1974) estimates for the rate of convergence in the Lindeberg theorem and also compute the appearing absolute constants. We introduce the asymptotically exact constants in the constructed inequalities and obtain upper bounds for them. We analyze the values of Esseen's, Rozovskii's, and Lyapunov's fractions, compare them pairwise and provide some extremal distributions. As an auxiliary statement, we prove a sharp inequality for the quadratic tails of an arbitrary distribution (with finite second order moment) and its convolutional symmetrization.