Stein's method, heat kernel, and linear functions on the orthogonal groups

TL;DR

This paper combines Stein's method with heat kernel techniques to analyze the distribution of linear functions on orthogonal groups, providing improved bounds on their approximation to normal distribution.

Contribution

It introduces a novel combination of Stein's method and heat kernel techniques to bound the total variation distance for linear functions on orthogonal groups.

Findings

Bound on total variation distance improved to 2*sqrt(2)/(n-1)

Demonstrates effectiveness of combining Stein's method with heat kernel analysis

Provides tighter approximation bounds for linear functions on orthogonal groups

Abstract

Combining Stein's method with heat kernel techniques, we study the function Tr(AO), where A is a fixed n by n real matrix over such that Tr(AA^t)=n, and O is from the Haar measure of the orthogonal group O(n,R). It is shown that the total variation distance of the random variable Tr(AO) to a standard normal random variable is bounded by 2 * squareroot(2) /(n-1), slightly improving the constant in a bound of Meckes, which was obtained by completely different methods.