On Stein's method for multivariate normal approximation

TL;DR

This paper synthesizes two approaches to Stein's method for multivariate normal approximation, broadening its applicability and improving convergence rates, with applications to eigenfunctions on manifolds.

Contribution

It combines existing Stein's method techniques, extends their applicability, and demonstrates improved convergence rates with new and reworked applications.

Findings

Improved convergence rates in normal approximation.

Unified framework for discrete and continuous symmetries.

Application to eigenfunctions on Riemannian manifolds.

Abstract

The purpose of this paper is to synthesize the approaches taken by Chatterjee-Meckes and Reinert-R\"ollin in adapting Stein's method of exchangeable pairs for multivariate normal approximation. The more general linear regression condition of Reinert-R\"ollin allows for wider applicability of the method, while the method of bounding the solution of the Stein equation due to Chatterjee-Meckes allows for improved convergence rates. Two abstract normal approximation theorems are proved, one for use when the underlying symmetries of the random variables are discrete, and one for use in contexts in which continuous symmetry groups are present. The application to runs on the line from Reinert-R\"ollin is reworked to demonstrate the improvement in convergence rates, and a new application to joint value distributions of eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian manifold is presented.