Analysis of The Limiting Spectral Distribution of Large Random Matrices of The Mar\v{c}enko-Pastur Type

Abstract

Consider the random matrix \(\bW_n = \bB_n + n^{-1}\bX_n^*\bA_n\bX_n\), where \(\bA_n\) and \(\bB_n\) are Hermitian matrices of dimensions \(p \times p\) and \(n \times n\), respectively, and \(\bX_n\) 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 \(\bA_n\) and \(\bB_n\) converge as \(p, n \to \infty\) towards two probability measures \(\calA\) and \(\calB\). Building on the groundbreaking work of \cite{marchenko1967distribution}, which demonstrated that the empirical spectral distribution of \(\bW_n\) converges towards a probability measure \(F\) characterized by its Stieltjes transform, this paper investigates the properties of \(F\) when \(\calB\) is a general measure. We show that \(F\) has an analytic density at the region near where the Stieltjes transform of $\calB$ is bounded. The density closely resembles \(C\sqrt{|x - x_0|}\) near certain edge points \(x_0\) of its support for a wide class of \(\calA\) and \(\calB\). We provide a complete characterization of the support of \(F\). Moreover, we show that \(F\) can exhibit discontinuities at points where \(\calB\) is discontinuous.