Analysis of the limiting spectral distribution of large random matrices of the Marčenko–Pastur type

Consider the random matrix [Formula: see text], where [Formula: see text] and [Formula: see text] are Hermitian matrices of dimensions [Formula: see text] and [Formula: see text], respectively, and [Formula: see text] is a [Formula: see text] random matrix with independent and identically distributed entries of mean 0 and variance 1. Assume that [Formula: see text] and [Formula: see text] grow to infinity proportionally, and that the spectral measures of [Formula: see text] and [Formula: see text] converge as [Formula: see text] towards two probability measures [Formula: see text] and [Formula: see text]. Building on the groundbreaking work of [V. A. Marčenko and L. A. Pastur, Distribution of eigenvalues for some sets of random matrices, Mat. Sb. 114(4) (1967) 507–536], which demonstrated that the empirical spectral distribution of [Formula: see text] converges towards a probability measure [Formula: see text] characterized by its Stieltjes transform, this paper investigates the properties of [Formula: see text] when [Formula: see text] is a general measure. We show that [Formula: see text] has an analytic density at the region near where the Stieltjes transform of [Formula: see text] is bounded. The density closely resembles [Formula: see text] near certain edge points [Formula: see text] of its support for a wide class of [Formula: see text] and [Formula: see text]. We provide a complete characterization of the support of [Formula: see text]. Moreover, we show that [Formula: see text] can exhibit discontinuities at points where [Formula: see text] is discontinuous.