Alexey Naumov

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 Wigners 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.

Distribution of Linear Statistics of Singular Values of the Product of Random Matrices

In this paper we consider the product of two independent random matrices

On minimal singular values of random matrices with correlated entries

\mathbb X^{(1)}

Asymptotic Analysis of Symmetric Functions

and

Local laws for non-Hermitian random matrices

\mathbb X^{(2)}

Local semicircle law under weak moment conditions

. Assume that

The strong law of large numbers for random processes

X_{jk}^{(q)}, 1 \le j,k \le n, q = 1, 2,

ELLIPTIC LAW FOR REAL RANDOM MATRICES

are i.i.d. random variables with

Semicircle Law for a Class of Random Matrices with Dependent Entries

\mathbb E X_{jk}^{(q)} = 0, \mathbb E (X_{jk}^{(q)})^2 = 1

On the local semicircular law for Wigner ensembles

. Denote by