Limit Theorems for Two Classes of Random Matrices with Gaussian Entries

Abstract

In this note, we consider ensembles of random symmetric matrices with Gaussian elements. Assume that $$ \mathbb{E} $$ X ij = 0 and $$ \mathbb{E}{X}_{ij}^2={\sigma}_{ij}^2 $$ We do not assume that all the σ ij are equal. Assuming that the average of the normalized sums of variances in each row converges to one and the Lindeberg condition holds, we prove that the empirical spectral distribution of eigenvalues converges to Wigner’s semicircle law. We also provide an analogue of this result for sample covariance matrices and prove the convergence to the Marchenko–Pastur law. Bibliography: 5 titles.