Convergence rates to the Marchenko-Pastur type distribution

Abstract

S n = 1 n T 1/2 n X nX n T 1/2 n , where X n = (xi j ) is a p n matrix consisting of independent complex entries with mean zero and variance one, Tn is a p p nonrandom positive definite Hermitian matrix with spectral norm uniformly bounded in p. In this paper, if supn supi, j E | x 8 i j |> ∞ and y n = p/n > 1 uniformly as n → ∞, we obtain that the rate of the expected empirical spectral distribution of Sn converging to its limit spectral distribution is O(n 1/2). Moreover, under the same assumption, we prove that for any < 0, the rates of the convergence of the empirical spectral distribution of Sn in probability and the almost sure convergence are O(n 2/5) and O(n 2/5+) respectively. © 2011 Elsevier B.V. All rights reserved.