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

A. Pajor ¹   ; L. Pastur ²

¹ Department of Mathematics University Paris-Est Marne-la-Vallée, France
² Theoretical Department Institute for Low Temperatures Kharkiv, Ukraine

Studia Mathematica, Tome 195 (2009) no. 1, pp. 11-29

Résumé

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.

Détail

DOI : 10.4064/sm195-1-2

Keywords: consider times real symmetric hermitian random matrices sums non random matrix rank one matrices determined isotropic random vectors log concave probability law real amplitudes analog setting marchenko pastur mat prove rightarrow infty rightarrow infty distribution eigenvalues distribution amplitudes converge weakly distribution eigenvalues converges weakly probability non random limit found marchenko pastur