NORMAL LIMIT-THEOREMS FOR SYMMETRICAL RANDOM MATRICES

Using the machinery of zonal polynomials, we examine the limiting beha vior of random symmetric matrices invariant under conjugation by ortho gonal matrices as the dimension tends to infinity. In particular, we g ive sufficient conditions for the distribution of a fixed submatrix to tend to a normal distribution. We also consider the problem of when t he sequence of partial sums of the diagonal elements tends to a Browni an motion. Using these results, we show that if O-n is a uniform rando m n x n orthogonal matrix, then for any fixed k > 0, the sequence of p artial sums of the diagonal of O-n(k) tends to a Brownian motion as n --> infinity.