Total variation approximation of random orthogonal matrices by Gaussian matrices

TL;DR

This paper investigates how the distribution of sub-blocks of Haar-distributed orthogonal matrices approaches Gaussian distribution in total variation distance as matrix size grows, establishing convergence under certain conditions.

Contribution

It provides a rigorous analysis of the total variation approximation of Haar orthogonal matrices by Gaussian matrices, revealing conditions for convergence.

Findings

Total variation distance converges to zero when p_n q_n = o(n).

Sub-blocks of Haar orthogonal matrices can be approximated by Gaussian variables.

The result quantifies the asymptotic behavior of random orthogonal matrices.

Abstract

The topic of this paper is the asymptotic distribution of random orthogonal matrices distributed according to Haar measure. We examine the total variation distance between the joint distribution of the entries of $W_n$, the $p_n \times q_n$ upper-left block of a Haar-distributed matrix, and that of $p_nq_n$ independent standard Gaussian random variables. We show that the total variation distance converges to zero when $p_nq_n = o(n)$.