Extreme Eigenvalues of Large Dimensional Quaternion Sample Covariance Matrix

Abstract

In this paper, we shall investigate the almost sure limits of the largest and smallest eigenvalues of a quaternion sample covariance matrix. Suppose that ${\mathbf{X}}_n$ is a $p\times n$ matrix whose elements are independent quaternion variables with mean zero, variance 1 and uniformly bounded fourth moments. Denote ${\mathbf{S}}_n=\frac{1}{n}{\mathbf{X}}_n{\mathbf{X}}_n^{\ast }$. In this paper, we shall show that $s_{\max }\left({\mathbf{S}}_n\right)=s_p\left({\mathbf{S}}_n\right)\to {\left(1+\sqrt{y}\right)}^2,a.s.$ and $s_{\min }\left({\mathbf{S}}_n\right)\to {\left(1-\sqrt{y}\right)}^2,a.s.$ as $n\to \infty$, where $y=\lim p/n$, $s_1\left({\mathbf{S}}_n\right)\le \cdots \le s_p\left({\mathbf{S}}_n\right)$ are the eigenvalues of ${\mathbf{S}}_n$, $s_{\min }\left({\mathbf{S}}_n\right)=s_{p-n+1}\left({\mathbf{S}}_n\right)$ when $p>n$ and $s_{\min }\left({\mathbf{S}}_n\right)=s_1\left({\mathbf{S}}_n\right)$ when $p\le n$. We also prove that the set of conditions are necessary for $s_{\max }\left({\mathbf{S}}_n\right)\to {\left(1+\sqrt{y}\right)}^2,a.s.$ when the entries of ${\mathbf{X}}_n$ are i. i. d.

21 pages

References in corpus (2)

• On the limit of extreme eigenvalues of large dimensional random quaternion matrices
• Convergence of Empirical Spectral Distributions of Large Dimensional Quaternion Sample Covariance Matrices

Cited by in corpus (1)

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