Extreme eigenvalues of large dimensional quaternion sample covariance matrices

•Derived the strong limits of largest and smallest eigenvalues of large dimensional quaternion sample covariance matrices.•Modified the graphic theory for random complex matrices to that of random quaternion matrices.•Modified the Hadamard block product and diamond product of matrices to quaternion matrices.

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×np×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→∞n→∞, where y=limp/ny=limp/n, s1(Sn)≤⋯≤sp(Sn) are the eigenvalues of Sn, smin(Sn)=sp−n+1(Sn) when p>np>n and smin(Sn)=s1(Sn) when p≤np≤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.