Total Variation Approximation of Random Orthogonal Matrices by Gaussian Matrices

Abstract

The topic of this paper is the asymptotic distribution of the entries 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 and show that the total variation distance converges to zero when $$p_nq_n = o(n)$$.