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

In this paper, the authors show that the smallest (if p ≤ 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 - √ y)² as n → ∞ and p/n → y ∈ (0,∞), where X is a p × 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 + √ 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.