LIMIT OF THE SMALLEST EIGENVALUE OF A LARGE DIMENSIONAL SAMPLE COVARIANCE-MATRIX

In this paper, the authors show that the smallest (if p less-than-or-equal-to 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 - square-root y)2 as n and p/n --> y is-an-element-of (0, infinity), where X is a p x 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 + square-root 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.