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Normal limit theorems for symmetric random matrices

Rains, Eric M. (1998) Normal limit theorems for symmetric random matrices. Probability Theory and Related Fields, 112 (3). pp. 411-423. ISSN 0178-8051. doi:10.1007/s004400050195.

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Using the machinery of zonal polynomials, we examine the limiting behavior of random symmetric matrices invariant under conjugation by orthogonal matrices as the dimension tends to infinity. In particular, we give sufficient conditions for the distribution of a fixed submatrix to tend to a normal distribution. We also consider the problem of when the sequence of partial sums of the diagonal elements tends to a Brownian motion. Using these results, we show that if O_n is a uniform random n×n orthogonal matrix, then for any fixed k>0, the sequence of partial sums of the diagonal of O^k_n tends to a Brownian motion as n→∞.

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Additional Information:© 1998 Springer-Verlag Berlin Heidelberg. Received: 3 February 1998 ;  Revised version: 11 June 1998. The author would like to thank P. Diaconis for suggesting the problems of Theorem 3.2. The author would also like to thank the anonymous referee for pointing out a flaw in the convergence conditions in an earlier draft, as well as J. Lagarias for related helpful discussions, including the proof of Lemma 0.1.
Issue or Number:3
Classification Code:Mathematics Subject Classification (1991): Primary 15A52; Secondary 60F05
Record Number:CaltechAUTHORS:20171107-075613066
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Official Citation:Rains, E. Probab Theory Relat Fields (1998) 112: 411.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:83016
Deposited By: Tony Diaz
Deposited On:07 Nov 2017 18:59
Last Modified:15 Nov 2021 19:54

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