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

TL;DR

This paper investigates the convergence rates of the empirical spectral distribution of large quaternion sample covariance matrices, establishing precise bounds for expected, weak, and strong convergence under certain conditions.

Contribution

It provides the first detailed analysis of convergence rates for the spectral distribution of large quaternion covariance matrices, extending classical results to quaternion entries.

Findings

Expected ESD converges at rate O(n^{-1/2}a_n^{-3/4}) or O(n^{-1/5}) depending on a_n.

Weak convergence rate of ESD is O(n^{-2/5}a_n^{-2/5}) or O(n^{-1/5}).

Strong convergence rate of ESD is O(n^{-2/5+ exteta}a_n^{-2/5}) or O(n^{-1/5}).

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 $\eta>0$.