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2012 Tracy-Widom law for the extreme eigenvalues of sample correlation matrices
Zhigang Bao, Guangming Pan, Wang Zhou
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Electron. J. Probab. 17: 1-32 (2012). DOI: 10.1214/EJP.v17-1962

Abstract

Let the sample correlation matrix be $W=YY^T$ where $Y=(y_{ij})_{p,n}$ with $y_{ij}=x_{ij}/\sqrt{\sum_{j=1}^nx_{ij}^2}$. We assume $\{x_{ij}: 1\leq i\leq p, 1\leq j\leq n\}$ to be a collection of independent symmetrically distributed random variables with sub-exponential tails. Moreover, for any $i$, we assume $x_{ij}, 1\leq j\leq n$ to be identically distributed. We assume $0 \lt p \lt n$ and $p/n\rightarrow y$ with some $y\in(0,1)$ as $p,n\rightarrow\infty$. In this paper, we provide the Tracy-Widom law ($TW_1$) for both the largest and smallest eigenvalues of $W$. If $x_{ij}$ are i.i.d. standard normal, we can derive the $TW_1$ for both the largest and smallest eigenvalues of the matrix $\mathcal{R}=RR^T$, where $R=(r_{ij})_{p,n}$ with $r_{ij}=(x_{ij}-\bar x_i)/\sqrt{\sum_{j=1}^n(x_{ij}-\bar x_i)^2}$, $\bar x_i=n^{-1}\sum_{j=1}^nx_{ij}$.

Citation

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Zhigang Bao. Guangming Pan. Wang Zhou. "Tracy-Widom law for the extreme eigenvalues of sample correlation matrices." Electron. J. Probab. 17 1 - 32, 2012. https://doi.org/10.1214/EJP.v17-1962

Information

Accepted: 4 October 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1254.15036
MathSciNet: MR2988403
Digital Object Identifier: 10.1214/EJP.v17-1962

Subjects:
Primary: 15B52
Secondary: 62H10, 62H25

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Vol.17 • 2012
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