We consider n × n real symmetric and hermitian random matrices Hₙ that are sums of a non-random matrix 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 and the distribution of amplitudes converge weakly, then the distribution of eigenvalues of Hₙ converges weakly in probability to the non-random limit, found by Marchenko and Pastur.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm195-1-2,
author = {A. Pajor and L. Pastur},
title = {On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution},
journal = {Studia Mathematica},
volume = {192},
year = {2009},
pages = {11-29},
zbl = {1178.15023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm195-1-2}
}
A. Pajor; L. Pastur. On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution. Studia Mathematica, Tome 192 (2009) pp. 11-29. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm195-1-2/