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

A. Pajor, and L. Pastur. "On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution." Studia Mathematica 195.1 (2009): 11-29. <http://eudml.org/doc/284873>.

@article{A2009,
abstract = {We consider n × n real symmetric and hermitian random matrices Hₙ 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 $Hₙ^\{(0)\}$ 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.},
author = {A. Pajor, L. Pastur},
journal = {Studia Mathematica},
keywords = {eigenvalue distribution; log-concave measures; asymptotic theory of convex bodies; Hermitian random matrices},
language = {eng},
number = {1},
pages = {11-29},
title = {On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution},
url = {http://eudml.org/doc/284873},
volume = {195},
year = {2009},
}

TY - JOUR
AU - A. Pajor
AU - L. Pastur
TI - On the limiting empirical measure of eigenvalues of the sum of rank one matrices with log-concave distribution
JO - Studia Mathematica
PY - 2009
VL - 195
IS - 1
SP - 11
EP - 29
AB - We consider n × n real symmetric and hermitian random matrices Hₙ 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 $Hₙ^{(0)}$ 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.
LA - eng
KW - eigenvalue distribution; log-concave measures; asymptotic theory of convex bodies; Hermitian random matrices
UR - http://eudml.org/doc/284873
ER -