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

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Pajor A., Pastur L.

Studia Mathematica

2009

tom 195

nr 1

11-29

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.

Ograniczanie wyników

1 Studia Mathematica

1 Pajor A.

1 Pastur L.

1 2009

Wyniki wyszukiwania

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