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

• Autorzy: A. Pajor , L. Pastur
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 52A23: Asymptotic theory of convex bodies
• 47A75: Eigenvalue problems
• 15B52: Random matrices

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2009
• Tom: 195
• Numer: 1
• Strony: 11-29

Daty

wydano

2009

Twórcy

autor

A. Pajor

• Department of Mathematics, University Paris-Est, Marne-la-Vallée, France

autor

L. Pastur

• Theoretical Department, Institute for Low Temperatures, Kharkiv, Ukraine

Identyfikatory

DOI

10.4064/sm195-1-2