In the first part of the paper, we develop certain inequalities to bound the difference between distributions in terms of their Stieltjes transforms and established a convergence rate of expected spectral distributions of large Wigner matrices. The second part is devoted to establishing convergence rates for the sample covariance matrices, for the cases where the ratio of the dimension to the degrees of freedom is bounded away from 1 or close to 1, respectively.
Z. D. Bai. "Convergence Rate of Expected Spectral Distributions of Large Random Matrices. Part II. Sample Covariance Matrices." Ann. Probab. 21 (2) 649 - 672, April, 1993. https://doi.org/10.1214/aop/1176989262