Multivariate normal approximation using exchangeable pairs

TL;DR

This paper develops a multivariate extension of Stein's exchangeable pairs method, providing new theorems and applications for normal approximation in high-dimensional group settings.

Contribution

It introduces three abstract theorems for multivariate normal approximation using exchangeable pairs, filling a gap in Stein's method for multivariate cases.

Findings

Proved approximate normality of rank k projections of Haar measure on orthogonal groups

Extended exchangeable pairs method to continuous symmetry groups

Established theoretical bounds for multivariate normal approximation

Abstract

Since the introduction of Stein's method in the early 1970s, much research has been done in extending and strengthening it; however, there does not exist a version of Stein's original method of exchangeable pairs for multivariate normal approximation. The aim of this article is to fill this void. We present three abstract normal approximation theorems using exchangeable pairs in multivariate contexts, one for situations in which the underlying symmetries are discrete, and real and complex versions of a theorem for situations involving continuous symmetry groups. Our main applications are proofs of the approximate normality of rank $k$ projections of Haar measure on the orthogonal and unitary groups, when $k=o(n)$.