Limit of the smallest eigenvalue of a large dimensional sample covariance matrix. (English) Zbl 0779.60026

Bai, Z. D.; Yin, Y. Q.

Ann. Probab. 21, No. 3, 1275-1294 (1993).

Summary: 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 \(n\to\infty\) and \(p/n\to y\in(0,\infty)\), where \(X\) is a \(p\times n\) matrix with i.i.d. 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 the authors and P. R. Krishnaiah [Probab. Theory Relat. Fields 78, No. 4, 509–521 (1988; Zbl 0627.62022)]. The present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices.

MSC:

60F15	Strong limit theorems
62H99	Multivariate analysis

Keywords:

random matrix; sample covariance matrix; smallest eigenvalue of a random matrix; spectral radius

Citations:

Zbl 0641.62015; Zbl 0627.62022