Rate of convergence in probability to the Marchenko-Pastur law

Abstract

It is shown that the Kolmogorov distance between the spectral distribution function of a random covariance matrix (1/p)XX^T, where X is an n×p matrix with independent entries and the distribution function of the Marchenko-Pastur law is of order O(n^-1/2) in probability. The bound is explicit and requires that the twelfth moment of the entries of the matrix is uniformly bounded and that p/n is separated from 1.