Limiting Eigenvalue Behavior of a Class of Large Dimensional Random Matrices Formed From a Hadamard Product

Abstract

This paper investigates the strong limiting behavior of the eigenvalues of the class of matrices 1 N (Dn ◦Xn)(Dn ◦Xn)∗, studied in Girko 2001. Here, Xn = (xij) is an n×N random matrix consisting of independent complex standardized random variables, Dn = (dij), n × N , has nonnegative entries, and ◦ denotes Hadamard (componentwise) product. Results are obtained under assumptions on the entries of Xn and Dn which are different from those in Girko (2001), which include a Lindeberg condition on the entries of Dn ◦Xn, as well as a bound on the average of the rows and columns of Dn ◦ Dn. The present paper separates the assumptions needed on Xn and Dn. It assumes a Lindeberg condition on the entries of Xn, along with a tigntness-like condition on the entries of Dn,