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

TL;DR

This paper studies the eigenvalue distribution of large random matrices formed by the Hadamard product of a deterministic matrix and a random matrix, extending Girko's 2001 results with new assumptions and separating conditions on the matrices.

Contribution

It introduces new assumptions on the matrices involved, including a Lindeberg condition on the random matrix and a tightness condition on the deterministic matrix, advancing the understanding of eigenvalue behavior.

Findings

Eigenvalue distribution converges under new assumptions.

Separated conditions on random and deterministic matrices.

Extended results beyond Girko's original framework.

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 $D_n$,