Extreme eigenvalues of large dimensional quaternion sample covariance matrices

作者：Li, Huiqin; Bai, Zhidong^*

来源：Journal of Statistical Planning and Inference, 2015, 159: 1-14.

DOI：10.1016/j.jspi.2014.10.005

• Extreme eigenvalues
• Large dimension
• Quaternion matrices
• Random matrix theory
• Sample covariance matrix

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 x n matrix whose elements are independent quaternion variables with mean zero, variance 1 and uniformly bounded fourth moments. Denote S-n = 1/nX(n)X(n)*. In this paper, we shall show that s(max) (S-n) = s(p) (S-n) -> (1 + root y)(2), a.s. and s(min) (S-n) -> (1 + root y)(2), a.s. and s(min) (S-n) -> (1 - root y)(2) , a.s. as n -> infinity, 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 + root y)(2), a.s. when the entries of X-n are i. i. d.

• 出版日期2015-4
• 单位东北师范大学