Extreme eigenvalues of large dimensional quaternion sample covariance matrices

Abstract

In this paper, we investigate the almost sure limits of the largest and smallest eigenvalues of a quaternion sample covariance matrix. Suppose that Xn is a p×n matrix whose elements are independent quaternion variables with mean zero, variance 1 and uniformly bounded fourth moments. Denote Sn=1nXnXn∗. In this paper, we shall show that smax(Sn)=sp(Sn)→(1+y)2,a.s. and smin(Sn)→(1−y)2,a.s. as n→∞, where y=limp/n, s1(Sn)≤⋯≤sp(Sn) are the eigenvalues of Sn, smin(Sn)=sp−n+1(Sn) when p>n and smin(Sn)=s1(Sn) when p≤n. We also prove that the set of conditions are necessary for smax(Sn)→(1+y)2,a.s. when the entries of Xn are i. i. d.