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 + R1=2XTXHR1=2 , where A, R and T are Hermitian nonnegative deÖnite matrices, such that R and T have bounded spectral norm with T being diagonal, and R1=2 is the nonnegative deÖnite 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 with bounded trace norm and for each complex z outside the positive real line, Tr h (B zIM) 1 i M (z) ! 0 almost surely as M; N ! 1 at the same rate, where M (z) is deterministic and solely depends on ; 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 Öeld of statistical signal processi