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→∞.
Article PDF
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: 3 February 1998 / Revised version: 11 June 1998
Rights and permissions
About this article
Cite this article
Rains, E. Normal limit theorems for symmetric random matrices. Probab Theory Relat Fields 112, 411–423 (1998). https://doi.org/10.1007/s004400050195
Issue Date:
DOI: https://doi.org/10.1007/s004400050195