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

We consider n × n real symmetric and hermitian random matrices Hn that are sums of a non-random matrix H n 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 H n 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.

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