Limiting eigenvalue behavior of a class of large dimensional random matrices formed from a Hadamard product

This paper investigates the strong limiting behavior of the eigenvalues of the class of matrices 1/N (D-n circle X-n)(D-n circle X-n)*, studied in [V. L. Girko, Theory of Stochastic Canonical Equations: Vol. 1 (Kluwer Academic Publishers, Dordrecht, 2001)]. Here, X-n = (x(ij)) is an n x N random matrix consisting of independent complex standardized random variables, D-n = (d(ij)), n x N, has nonnegative entries, and. denotes Hadamard (componentwise) product. Results are obtained under assumptions on the entries of X-n and D-n which are different from those in [V. L. Girko, Theory of Stochastic Canonical Equations: Vol. 1 (Kluwer Academic Publishers, Dordrecht, 2001)], which include a Lindeberg condition on the entries of D-n circle X-n, as well as a bound on the average of the rows and columns of D-n circle D-n. The present paper separates the assumptions needed on X-n and D-n. It assumes a Lindeberg condition on the entries of X-n, along with a tightness-like condition on the entries of D-n.