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

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