Convergence Rates of ESD

Abstract

In applications of asymptotic theorems of spectral analysis of large dimensional random matrices, one of the important problems is the convergence rate of the ESD. It had been puzzling probabilists for a long time until the papers of Bai [16, 17] were published. The moment approach to establishing limiting theorems for spectral analysis of large dimensional random matrices is to show that each moment of the ESD tends to a nonrandom limit. This proves the existence of the LSD by applying the Carleman criterion. This method successfully established the existence of the LSD of large dimensional Wigner matrices, sample covariance matrices, and multivariate F-matrices. However, this method cannot give any convergence rate. In Bai [16], three inequalities were established in terms of the difference of Stieltjes transforms (see Chapter B). In this chapter, we shall apply these inequalities to establish the convergence rates for the ESD of largeWigner and sample covariance matrices.