## The Annals of Probability

### 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.

#### Article information

Source
Ann. Probab., Volume 21, Number 3 (1993), 1275-1294.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176989118

Digital Object Identifier
doi:10.1214/aop/1176989118

Mathematical Reviews number (MathSciNet)
MR1235416

Zentralblatt MATH identifier
0779.60026

JSTOR