Analysis of The Limiting Spectral Distribution of Large Random Matrices of The Marčenko-Pastur Type

Consider the random matrix ${\text{bW}}_n={\text{bB}}_n+n^{-1}{\text{bX}}_n^{\ast }{\text{bA}}_n{\text{bX}}_n$, where ${\text{bA}}_n$ and ${\text{bB}}_n$ are Hermitian matrices of dimensions $p\times p$ and $n\times n$, respectively, and ${\text{bX}}_n$ is a $p\times n$ random matrix with independent and identically distributed entries of mean 0 and variance 1. Assume that $p$ and $n$ grow to infinity proportionally, and that the spectral measures of ${\text{bA}}_n$ and ${\text{bB}}_n$ converge as $p,n\to \mathrm{\infty }$ towards two probability measures $\text{calA}$ and $\text{calB}$. Building on the groundbreaking work of \cite{marchenko1967distribution}, which demonstrated that the empirical spectral distribution of ${\text{bW}}_n$ converges towards a probability measure $F$ characterized by its Stieltjes transform, this paper investigates the properties of $F$ when $\text{calB}$ is a general measure. We show that $F$ has an analytic density at the region near where the Stieltjes transform of $\text{calB}$ is bounded. The density closely resembles $C\sqrt{\mathrm{\mid }x-x_0\mathrm{\mid }}$ near certain edge points $x_0$ of its support for a wide class of $\text{calA}$ and $\text{calB}$. We provide a complete characterization of the support of $F$. Moreover, we show that $F$ can exhibit discontinuities at points where $\text{calB}$ is discontinuous.