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

TL;DR

This paper studies the spectral distribution of large random matrices of the Marčenko-Pastur type, revealing detailed properties of the limiting spectral measure, including density behavior, support, and discontinuities, extending classical results to more general settings.

Contribution

It provides a comprehensive analysis of the limiting spectral distribution for a broad class of large random matrices with general spectral measures, including support characterization and density behavior near edges.

Findings

The limiting spectral distribution has an analytic density near regions where the Stieltjes transform of B is bounded.

The density near support edges behaves like a square root function, i.e., proportional to a0|x - x_0|.

The spectral measure can have discontinuities at points where the measure B is discontinuous.

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.