On the Limiting Empirical Measure of the sum of rank one matrices with log-concave distribution

Abstract

We consider $n\times n$ real symmetric and hermitian random matrices $H_{n,m}$ equals the sum of a non-random matrix $H_n^{(0)}$ matrix and the sum of $m$ rank-one matrices determined by $m$ i.i.d. isotropic random vectors with log-concave probability law and i.i.d. random amplitudes $\{{\tau }_{\alpha }{\}}_{\alpha =1}^m$. This is a generalization of the case of vectors uniformly distributed over the unit sphere, studied in [Marchenko-Pastur (1967)]. We prove that if $n\to \infty ,m\to \infty ,m/n\to c\in [0,\infty )$ and that the empirical eigenvalue measure of $H_n^{(0)}$ converges weakly, then the empirical eigenvalue measure of $H_{n,m}$ converges in probability to a non-random limit, found in [Marchenko-Pastur (1967)].