Convergence rates of spectral distributions of large dimensional quaternion sample covariance matrices

In this paper, we study the convergence rates of empirical spectral distributions of large dimensional quaternion sample covariance matrices. Assume that the entries of Xn (p×np×n) are independent quaternion random variables with means zero, variances 1 and uniformly bounded sixth moments. Denote Sn=1nXnXn∗. Using Bai’s inequality, we prove that the expected empirical spectral distribution (ESD) converges to the limiting Marčenko–Pastur distribution with the ratio of dimension to sample size yp=p/nyp=p/n at a rate of O(n−1/2an−3/4) when an>n−2/5an>n−2/5 or O(n−1/5)O(n−1/5) when an≤n−2/5an≤n−2/5, where an=(1−yp)2. Moreover, the rates for both the convergence in probability and the almost sure convergence are also established. The weak convergence rate of the ESD is O(n−2/5an−1/2) when an>n−2/5an>n−2/5 or O(n−1/5)O(n−1/5) when an≤n−2/5an≤n−2/5. The strong convergence rate of the ESD is O(n−2/5+ηan−1/2) when an>κn−2/5an>κn−2/5 or O(n−1/5)O(n−1/5) when an≤κn−2/5an≤κn−2/5 for any η>0η>0 where κκ is a positive constant.