Normal limit theorems for symmetric random matrices

Creators: Rains, Eric M.

Abstract

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→∞.

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.

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Normal limit theorems for symmetric random matrices

Abstract

Additional Information

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