Variations on the Berry-Esseen theorem

TL;DR

This paper investigates the accuracy of Gaussian approximations for sums of independent random variables, revealing universal combinations that outperform traditional sums and extending analysis to non-identically distributed variables.

Contribution

It introduces universal linear combinations that improve Gaussian approximation quality and extends the analysis to independent, non-identically distributed variables.

Findings

Universal linear combinations outperform sums in Gaussian approximation.

Improved bounds for sums of i.i.d. variables with finite fourth moments.

Extension of results to independent, non-i.i.d. variables.

Abstract

We analyze the quality of the gaussian approximation to linear combinations of n independent, identically-distributed random variables with finite fourth moments. It turns out that there exist universal, simple linear combinations that perform better than the sum of the variables. We also investigate the case in which the random variables are independent, yet they are not necessarily identically distributed.