Limit Theorems for Two Classes of Random Matrices with Dependent Entries

In this paper we study random symmetric matrices with dependent entries. Suppose that all entries have zero mean and finite variances, which can be different. Assuming that the average of normalized sums of variances in each row converges to one and the Lindeberg condition holds true, we prove that the empirical spectral distribution of eigenvalues converges to Wigner's semicircle law. The result can be generalized to the class of covariance matrices with dependent entries. In this case expected empirical spectral distribution function converges to the Marchenko--Pastur law.