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

Combining Stein's method with heat kernel techniques, we study the function Tr(AO), where A is a fixed n x n matrix over R such that Tr(AA(t)) = n, and Ois 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 root 2/n-1, slightly improving the constant in a bound of Meckes, which was obtained by completely different methods. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

By combining Stein's method with heat kernel techniques, this study investigates the properties of the function Tr(AO), where A is a fixed matrix and O is sampled from the Haar measure of the orthogonal group. The research shows that the total variation distance between Tr(AO) and a standard normal random variable is bounded by a certain constant, slightly improving upon previous results obtained using different methods.