Limit of the Smallest Eigenvalue of a Large Dimensional Sample Covariance Matrix

Abstract

In this paper, the authors show that the smallest (if $p \\leq n$) or the $(p - n + 1)$-th smallest (if $p > n$) eigenvalue of a sample covariance matrix of the form $(1/n)XX'$ tends almost surely to the limit $(1 - \\sqrt y)^2$ as $n \\rightarrow \\infty$ and $p/n \\rightarrow y \\in (0,\\infty)$, where $X$ is a $p \\times n$ matrix with iid entries with mean zero, variance 1 and fourth moment finite. Also, as a by-product, it is shown that the almost sure limit of the largest eigenvalue is $(1 + \\sqrt y)^2$, a known result obtained by Yin, Bai and Krishnaiah. The present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices.