### 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′ _{n}A _{n}) = 1. Then, as n → ∞, √n tr A _{n}H _{n} converges in distribution to the standard normal distribution.

Original language | English (US) |
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Pages (from-to) | 600-610 |

Number of pages | 11 |

Journal | Symmetry |

Volume | 3 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 2011 |

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### All Science Journal Classification (ASJC) codes

- Computer Science (miscellaneous)
- Chemistry (miscellaneous)
- Mathematics(all)
- Physics and Astronomy (miscellaneous)

### Cite this

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**High-dimensional random matrices from the classical matrix groups, and generalized hypergeometric functions of matrix argument.** / Richards, Donald St P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - High-dimensional random matrices from the classical matrix groups, and generalized hypergeometric functions of matrix argument

AU - Richards, Donald St P.

PY - 2011/9/1

Y1 - 2011/9/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84860552601&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860552601&partnerID=8YFLogxK

U2 - 10.3390/sym3030600

DO - 10.3390/sym3030600

M3 - Article

AN - SCOPUS:84860552601

VL - 3

SP - 600

EP - 610

JO - Symmetry

JF - Symmetry

SN - 2073-8994

IS - 3

ER -