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 $\frac1N(D_n\circ X_n)(D_n\circ X_n)^*$, studied in Girko 2001. Here, $X_n=(x_{ij})$ is an $n\times N$ random matrix consisting of independent complex standardized random variables, $D_n=(d_{ij})$, $n\times N$, has nonnegative entries, and $\circ$ denotes Hadamard (componentwise) product. Results are obtained under assumptions on the entries of $X_n$ and $D_n$ which are different from those in Girko (2001), which include a Lindeberg condition on the entries of $D_n\circ X_n$, as well as a bound on the average of the rows and columns of $D_n\circ D_n$. 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 tigntness-like condition on the entries of $D_n$,