High-dimensional random matrices from the classical matrix groups, and generalized hypergeometric functions of matrix argument

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.