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.
Keywords
MathematicsEigenvalues and eigenvectorsCombinatoricsLimit (mathematics)CovarianceCovariance matrixSample mean and sample covarianceMatrix (chemical analysis)Zero (linguistics)Central limit theoremRandom matrixMathematical analysisStatisticsPhysicsQuantum mechanics
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Publication Info
- Year
- 1993
- Type
- article
- Volume
- 21
- Issue
- 3
- Citations
- 532
- Access
- Closed
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Cite This
Zhidong Bai,
Yanqing Yin
(1993).
Limit of the Smallest Eigenvalue of a Large Dimensional Sample Covariance Matrix.
The Annals of Probability
, 21
(3)
.
https://doi.org/10.1214/aop/1176989118
Identifiers
- DOI
- 10.1214/aop/1176989118