On the Limiting Empirical Measure of the sum of rank one matrices with log-concave distribution
Résumé
We consider n × n real symmetric and hermitian random matrices Hn,m equals the sum of a non-random matrix H(0) matrix and the sum of m rank-one matrices determined by m i.i.d. isotropic random vectors with log-concave probability law and i.i.d. random amplitudes {τα}mα=1. This is a generalization of the case of vectors uniformly distributed over the unit sphere, studied in [17]. We prove as in [17] that if n → ∞, m → ∞, m/n → c ∈ [0,∞) and that the empirical eigenvalue measure of H(0) converges weakly, then the empirical eigenvalue measure of H converges in n n,m probability to a non-random limit, found in [17].