## The Annals of Probability

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

### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 62H99: None of the above, but in this section

**Keywords**

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

#### Citation

Bai, Z. D.; Yin, Y. Q. Limit of the Smallest Eigenvalue of a Large Dimensional Sample Covariance Matrix. Ann. Probab. 21 (1993), no. 3, 1275--1294. doi:10.1214/aop/1176989118. https://projecteuclid.org/euclid.aop/1176989118