Extreme eigenvalues of large dimensional quaternion sample covariance matrices

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