Spectral convergence for a general class of random matrices

Let X be an M x 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 Theta with bounded trace norm and for each complex z outside the positive real line, Tr [Theta (B - zI(M))(-1)] - delta(M) (z) -> 0 almost surely as M . N -> infinity at the same rate, where delta(M) (z) is deterministic and solely depends on Theta. 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. (C) 2011 Elsevier B.V. All rights reserved.