Abstract
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 $s \\rightarrow \\infty$.
Keywords
MathematicsWishart distributionEigenvalues and eigenvectorsCombinatoricsRandom matrixMatrix (chemical analysis)StatisticsMultivariate statisticsPhysics
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Publication Info
- Year
- 1985
- Type
- article
- Volume
- 13
- Issue
- 4
- Citations
- 254
- Access
- Closed
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Cite This
Jack W. Silverstein
(1985).
The Smallest Eigenvalue of a Large Dimensional Wishart Matrix.
The Annals of Probability
, 13
(4)
.
https://doi.org/10.1214/aop/1176992819
Identifiers
- DOI
- 10.1214/aop/1176992819