Spectral convergence for a general class of random matrices

Francisco Rubio , Xavier Mestre

Let be an complex random matrix with i.i.d. entries having mean zero and variance and consider the class of matrices of the type , where , and are Hermitian nonnegative definite matrices, such that and have bounded spectral norm with being diagonal, and is the nonnegative definite square-root of . Under some assumptions on the moments of the entries of , it is proved in this paper that, for any matrix with bounded trace norm and for each complex outside the positive real line, almost surely as at the same rate, where is deterministic and solely depends on and . The previous result can be particularized to the study of the limiting behavior of the Stieltjes transform as well as the eigenvectors of the random matrix model . The study is motivated by applications in the field of statistical signal processing.

Mots clés: Random matrix theory; Stieltjes transform; Multivariate statistics; Sample covariance matrix; Separable covariance model

Domaines: Bio-informatique [q-bio.QM]; Bio-Informatique, Biologie Systémique [q-bio.QM]

Francisco Rubio, Xavier Mestre. Spectral convergence for a general class of random matrices. Statistics and Probability Letters, 2011, 81 (5), pp.592. ⟨10.1016/j.spl.2011.01.004⟩. ⟨hal-00725102⟩