A New Method for Bounding Rates of Convergence of Empirical Spectral Distributions

The probabilistic properties of eigenvalues of random matrices whose dimension increases indefinitely has received considerable attention. One important aspect is the existence and identification of the limiting spectral distribution (LSD) of the empirical distribution of the eigenvalues. When the LSD exists, it is useful to know the rate at which the convergence holds. The main method to establish such rates is the use of Stieltjes transform. In this article we introduce a new technique of bounding the rates of convergence to the LSD. We show how our results apply to specific cases such as the Wigner matrix and the Sample Covariance matrix.