Bernoulli

  • Bernoulli
  • Volume 18, Number 4 (2012), 1405-1420.

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 and G.M. Pan

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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 $p/n\rightarrow0$.

Article information

Source
Bernoulli, Volume 18, Number 4 (2012), 1405-1420.

Dates
First available in Project Euclid: 12 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1352727817

Digital Object Identifier
doi:10.3150/11-BEJ381

Mathematical Reviews number (MathSciNet)
MR2995802

Zentralblatt MATH identifier
1279.60012

Keywords
empirical distribution maximum eigenvalue random matrices

Citation

Chen, B.B.; Pan, G.M. 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 (2012), no. 4, 1405--1420. doi:10.3150/11-BEJ381. https://projecteuclid.org/euclid.bj/1352727817


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