Approximate central limit theorems

TL;DR

This paper enhances the classical central limit theorem by providing bounds on various distances between distributions, using Stein's method, to establish approximate normality in broader, non-standard settings.

Contribution

It introduces refined asymptotic bounds on distributional distances, extending the classical CLT to cases where Lindeberg's condition is approximately satisfied.

Findings

Provides bounds on Kolmogorov, Wasserstein, and Prokhorov distances.

Establishes approximate CLTs for non-standard triangular arrays.

Utilizes Stein's method for deriving these bounds.

Abstract

We refine the classical Lindeberg-Feller central limit theorem by obtaining asymptotic bounds on the Kolmogorov distance, the Wasserstein distance, and the parametrized Prokhorov distances in terms of a Lindeberg index. We thus obtain more general approximate central limit theorems, which roughly state that the row-wise sums of a triangular array are approximately asymptotically normal if the array approximately satisfies Lindeberg's condition. This allows us to continue to provide information in non-standard settings in which the classical central limit theorem fails to hold. Stein's method plays a key role in the development of this theory.