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Symmetry 2011, 3(3), 600-610; doi:10.3390/sym3030600
Article

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

Department of Statistics, Pennsylvania State University, University Park, PA 16802-2111, USA
Received: 27 May 2011 / Revised: 16 August 2011 / Accepted: 23 August 2011 / Published: 26 August 2011
(This article belongs to the Special Issue Symmetry in Probability and Inference)
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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 Hn be uniformly distributed according to Haar measure on the group of n × n orthogonal matrices, and let An be a non-random n × n real matrix such that tr (A'nAn) = 1. Then, as n→∞, √n tr AnHn converges in distribution to the standard normal distribution.
Keywords: Generalized hypergeometric function of matrix argument; normal approximation; orthogonal matrix; random matrix; Stiefel manifold; symplectic matrix; unitary matrix; zonal polynomial Generalized hypergeometric function of matrix argument; normal approximation; orthogonal matrix; random matrix; Stiefel manifold; symplectic matrix; unitary matrix; zonal polynomial
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).
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Richards, D.S.P. High-Dimensional Random Matrices from the Classical Matrix Groups, and Generalized Hypergeometric Functions of Matrix Argument. Symmetry 2011, 3, 600-610.

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