On the norm and eigenvalue distribution of large random matrices

We study the eigenvalue distribution of NxN symmetric random matrices H-N(x , y) = N(-1/2)h(x, y), x, y = 1,..., N, where h(x, y), x less than or equal to y are Gaussian weakly dependent random variables. We prove that the nor malized eigenvalue counting function of H-N converges with probability 1 to a nonrandom function mu(lambda) as N --> infinity. We derive an equation f or the Stieltjes transform of the measure d mu(lambda) and show that the la tter has a compact support Lambda(mu). We find the upper bound for lim sup( N-->infinity)parallel to H(N)parallel to and study asymptotically the case when there are no eigenvalues of H-N outside of Lambda(mu) when N --> infin ity.