On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution
Volume 195 / 2009
Studia Mathematica 195 (2009), 11-29
MSC: 15B52, 52A23, 47A75.
DOI: 10.4064/sm195-1-2
Abstract
We consider $n\times n$ real symmetric and hermitian random matrices $H_{n}$ that are sums of a non-random matrix $H_{n}^{(0)}$ and of $m_{n}$ 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 $m_{n}/n\rightarrow c\in [0,\infty )$ as $n\rightarrow \infty $, and the distribution of eigenvalues of $H_{n}^{(0)}$ and 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.
