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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 which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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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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