The Annals of Probability

Limit of the Smallest Eigenvalue of a Large Dimensional Sample Covariance Matrix

Z. D. Bai and Y. Q. Yin

Full-text: Open access

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


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