An isometric study of the Lindeberg-Feller CLT via Stein's method

TL;DR

This paper employs Stein's method to generalize the Lindeberg-Feller CLT, establishing bounds on the Kolmogorov distance for sums of independent variables, with optimal and improved bounds demonstrated.

Contribution

It introduces a new bound for the Kolmogorov distance in the CLT using Stein's method, enhancing previous results with optimal and tighter bounds.

Findings

Upper bound of optimal order for the Kolmogorov distance

Lower bound that improves previous results

Demonstration of bounds through a natural example

Abstract

We use Stein's method to prove a generalization of the Lindeberg-Feller CLT providing an upper and a lower bound for the superior limit of the Kolmogorov distance between a normally distributed random variable and the rowwise sums of a rowwise independent triangular array of random variables which is asymptotically negligible in the sense of Feller. A natural example shows that the upper bound is of optimal order. The lower bound improves a result by Andrew Barbour and Peter Hall.