Probability Theory and Related Fields

, Volume 112, Issue 3, pp 411–423 | Cite as

Normal limit theorems for symmetric random matrices

  • Eric M. Rains
Article
  • 75 Downloads

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→∞.

Mathematics Subject Classification (1991): Primary 15A52; Secondary 60F05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Eric M. Rains
    • 1
  1. 1.AT&T Research, Room C290, 180 Park Avenue, Florham Park, NJ 07932-0971, USA. E-mail: rains@research.att.comUS

Personalised recommendations