On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution

Abstract

We consider n × n real symmetric and hermitian random matrices H n that are sums of a non-random matrix Hn(0) and of mn rank-one matrices determined by i.i.d. isotropic random vectors with log-concave probability law and real amplitudes. This is an analog of the setting of Marchenko and Pastur [Mat. Sb. 72 (1967)]. We prove that if mn/n → c ∈ [0,∞) as n → ∞, and the distribution of eigenvalues of Hn(0) the distribution of amplitudes converge weakly, then the distribution of eigenvalues of H n converges weakly in probability to the non-random limit, found by Marchenko and Pastur. © Instytut Matematyczny PAN, 2009.