On natural convergence rate estimates in the Lindeberg theorem

TL;DR

This paper introduces new estimates for the convergence rate in the Lindeberg theorem, generalizing previous inequalities and employing a novel proof technique based on extremal properties of certain fractions.

Contribution

It provides two new estimates involving algebraic third-order moments and the Lindeberg fraction, extending and unifying prior results with a novel proof approach.

Findings

New estimates for convergence rates in the Lindeberg theorem.

Generalization of classical inequalities with broader applicability.

A novel proof technique based on extremal properties.

Abstract

We prove two estimates of the rate of convergence in the Lindeberg theorem, involving algebraic truncated third-order moments and the classical Lindeberg fraction, which generalize a series of inequalities due to (Esseen, 1969), (Rozovskii, 1974), (Wang, Ahmad, 2016), some of our recent results in (Gabdullin, Makarenko, Shevtsova, 2018, 2019) and, up to constant factors, also (Katz, 1963), (Petrov, 1965), (Osipov, 1966). The technique used in the proof is completely different from that in (Wang, Ahmad, 2016) and is based on some extremal properties of introduced fractions which has not been noted in (Katz, 1963), (Petrov, 1965), (Wang, Ahmad, 2016).