On the Rate of Convergence to the Marchenko-Pastur Distribution

Let X = (Xjk) denote n×p random matrix with entries Xjk, which are independent for 1 ≤ j ≤ n,1 ≤ k ≤ p. We consider the rate of convergence of empirical spectral distribution function of matrix W = 1 XX ∗ to the Marchenko–Pastur law. We assume that EXjk = 0, EX 2 jk = 1 and that the distributions of the matrix elements Xjk have a uniformly sub exponential decay in the sense that there exists a constant { > 0 such that for any 1 ≤ j ≤ n, 1 ≤ k ≤ p and any t ≥ 1 we have Pr{|Xjk| > t} ≤ { −1 exp{−t { }. By means of a recursion argument it is shown that the Kolmogorov distance between the empirical spectral distribution of the sample covariance matrix W and the Marchenko–Pastur distribution is of order O(n −1 log b n) with some positive constant b > 0.