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November, 1985 The Smallest Eigenvalue of a Large Dimensional Wishart Matrix
Jack W. Silverstein
Ann. Probab. 13(4): 1364-1368 (November, 1985). DOI: 10.1214/aop/1176992819


For positive integers $s, n$ let $M_s = (1/s)V_sV^T_s$, where $V_s$ is an $n \times s$ matrix composed of i.i.d. $N(0, 1)$ random variables. Assume $n = n(s)$ and $n/s \rightarrow y \in (0, 1)$ as $s \rightarrow \infty$. Then it is shown that the smallest eigenvalue of $M_s$ converges almost surely to $(1 - \sqrt y)^2$ as $s \rightarrow \infty$.


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Jack W. Silverstein. "The Smallest Eigenvalue of a Large Dimensional Wishart Matrix." Ann. Probab. 13 (4) 1364 - 1368, November, 1985.


Published: November, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0591.60025
MathSciNet: MR806232
Digital Object Identifier: 10.1214/aop/1176992819

Primary: 60F15
Secondary: 62H99

Keywords: $\chi^2$ distribution , Gersgorin's theorem , Smallest eigenvalue of random matrix

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 4 • November, 1985
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