Convergence rates to the Marchenko–Pastur type distribution

Sn=1nTn1/2XnXn∗Tn1/2, where Xn=(xij) is a p×np×n matrix consisting of independent complex entries with mean zero and variance one, Tn is a p×pp×p nonrandom positive definite Hermitian matrix with spectral norm uniformly bounded in pp. In this paper, if supnsupi,jE∣xij8∣<∞ and yn=p/n<1yn=p/n<1 uniformly as n→∞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)O(n−1/2). Moreover, under the same assumption, we prove that for any η>0η>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)O(n−2/5) and O(n−2/5+η)O(n−2/5+η) respectively.