An Exposition of G\"otze's Estimation of the Rate of Convergence in the Multivariate Central Limit Theorem

TL;DR

This paper explains G"otze's approach to estimating the rate of convergence in the multivariate CLT using Stein's method, offering detailed derivations and revealing a different dimensional dependence of the constant.

Contribution

It provides a clear exposition of G"otze's main ideas and derivations, and uncovers a new dimensional dependence in the constant involved.

Findings

Different dimensional dependence of the constant compared to G"otze's original result

Detailed derivations of intermediate estimates in Stein's method

Enhanced understanding of convergence rates in multivariate CLT

Abstract

We provide an explanation of the main ideas underlying G\"otze's main result in using Stein's method. We also provide detailed derivations of various intermediate estimates. Curiously, we are led to a different dimensional dependence of the constant than that given G\"otze's paper. We would like to dedicate this to Charles Stein on the occasion of his 90th birthday.