Analysis of The Limiting Spectral Distribution of Large Random Matrices of The Marčenko-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.