Convergence Rates of Spectral Distribution of Large Dimensional Quaternion Sample Covariance Matrix

Abstract

In this paper, we study the convergence rates of empirical spectral distribution of large dimensional quaternion sample covariance matrix. Assume that the entries of $\mathbf X_n$ ($p\times n$) are independent quaternion random variables with mean zero, variance 1 and uniformly bounded sixth moments. Denote $\mathbf S_n=\frac{1}{n}\mathbf X_n\mathbf X_n^*$. Using Bai inequality, we prove that the expected empirical spectral distribution (ESD) converges to the limiting Mar${\rm \check{c}}$enko-Pastur distribution with the ratio of the dimension to sample size $y_p=p/n$ at a rate of $O\left(n^{-1/2}a_n^{-3/4}\right)$ when $a_n>n^{-2/5}$ or $O\left(n^{-1/5}\right)$ when $a_n\le n^{-2/5}$, where $a_n=(1-\sqrt{y_p})^2$ is the lower bound for the M-P law. 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 $O\left(n^{-2/5}a_n^{-2/5}\right)$ when $a_n>n^{-2/5}$ or $O\left(n^{-1/5}\right)$ when $a_n\le n^{-2/5}$. The strong convergence rate of the ESD is $O\left(n^{-2/5+\eta}a_n^{-2/5}\right)$ when $a_n>n^{-2/5}$ or $O\left(n^{-1/5}\right)$ when $a_n\le n^{-2/5}$ for any $\eta>0$.