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.