Spectral convergence for a general class of random matrices

Abstract

Let X be an M × N complex random matrix with i.i.d. entries having mean zero and variance 1 / N and consider the class of matrices of the type B = A + R 1 / 2 XTX H R 1 / 2 , where A , R and T are Hermitian nonnegative definite matrices, such that R and T have bounded spectral norm with T being diagonal, and R 1 / 2 is the nonnegative definite square root of R . Under some assumptions on the moments of the entries of X , it is proved in this paper that, for any matrix Θ with bounded trace norm and for each complex z outside the positive real line, Tr [ Θ ( B − z I M ) − 1 ] − δ M ( z ) → 0 almost surely as M , N → ∞ at the same rate, where δ M ( z ) is deterministic and solely depends on Θ , A , R and T . 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 B . The study is motivated by applications in the field of statistical signal processing.