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November 2012 Convergence of the largest eigenvalue of normalized sample covariance matrices when $p$ and $n$ both tend to infinity with their ratio converging to zero
B.B. Chen, G.M. Pan
Bernoulli 18(4): 1405-1420 (November 2012). DOI: 10.3150/11-BEJ381

Abstract

Let ${\mathbf{X}}_{p}=({\mathbf{s}}_{1},\ldots,{\mathbf{s}}_{n})=(X_{ij})_{p\times n}$ where $X_{ij}$’s are independent and identically distributed (i.i.d.) random variables with $EX_{11}=0$, $EX_{11}^{2}=1$ and $EX_{11}^{4}<\infty$. It is showed that the largest eigenvalue of the random matrix ${\mathbf{A}}_{p}=\frac{1}{2\sqrt{np}}({\mathbf{X}}_{p}{\mathbf{X}}_{p}^{\prime}-n{\mathbf{I}}_{p})$ tends to $1$ almost surely as $p\rightarrow\infty$, $n\rightarrow\infty$ with $p/n\rightarrow0$.

Citation

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B.B. Chen. G.M. Pan. "Convergence of the largest eigenvalue of normalized sample covariance matrices when $p$ and $n$ both tend to infinity with their ratio converging to zero." Bernoulli 18 (4) 1405 - 1420, November 2012. https://doi.org/10.3150/11-BEJ381

Information

Published: November 2012
First available in Project Euclid: 12 November 2012

zbMATH: 1279.60012
MathSciNet: MR2995802
Digital Object Identifier: 10.3150/11-BEJ381

Rights: Copyright © 2012 Bernoulli Society for Mathematical Statistics and Probability

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Vol.18 • No. 4 • November 2012
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