High-Dimensional Random Matrices from the Classical Matrix Groups, and Generalized Hypergeometric Functions of Matrix Argument

Abstract: Results from the theory of the generalized hypergeometric functions of matrix argument, and the related zonal polynomials, are used to develop a new approach to study the asymptotic distributions of linear functions of uniformly distributed random matrices from the classical compact matrix groups. In particular, we provide a new approach for proving the following result of D’Aristotile, Diaconis, and Newman: Let the random matrix H_n be uniformly distributed according to Haar measure on the group of n × n orthogonal matrices, and let A_n be a non-random n × n real matrix such that tr (A'_nA_n) = 1. Then, as n→∞, √n tr A_nH_n converges in distribution to the standard normal distribution.