Convergence rates to the Marchenko–Pastur type distribution

S n = 1 n T n 1 / 2 X n X n ∗ T n 1 / 2 , where X n = ( x i j ) is a p × n matrix consisting of independent complex entries with mean zero and variance one, T n is a p × p nonrandom positive definite Hermitian matrix with spectral norm uniformly bounded in p . In this paper, if sup n sup i , j E ∣ x i j 8 ∣ < ∞ and y n = p / n < 1 uniformly as n → ∞ , we obtain that the rate of the expected empirical spectral distribution of S n converging to its limit spectral distribution is O ( n − 1 / 2 ) . Moreover, under the same assumption, we prove that for any η > 0 , the rates of the convergence of the empirical spectral distribution of S n in probability and the almost sure convergence are O ( n − 2 / 5 ) and O ( n − 2 / 5 + η ) respectively.