## Electronic Journal of Probability

### Tracy-Widom law for the extreme eigenvalues of sample correlation matrices

#### 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 $r_{ij}=(x_{ij}-\bar x_i)/\sqrt{\sum_{j=1}^n(x_{ij}-\bar x_i)^2}$, $\bar x_i=n^{-1}\sum_{j=1}^nx_{ij}$.

#### Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 88, 32 pp.

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

https://projecteuclid.org/euclid.ejp/1465062410

Digital Object Identifier
doi:10.1214/EJP.v17-1962

Mathematical Reviews number (MathSciNet)
MR2988403

Zentralblatt MATH identifier
1254.15036

Rights

#### Citation

Bao, Zhigang; Pan, Guangming; Zhou, Wang. Tracy-Widom law for the extreme eigenvalues of sample correlation matrices. Electron. J. Probab. 17 (2012), paper no. 88, 32 pp. doi:10.1214/EJP.v17-1962. https://projecteuclid.org/euclid.ejp/1465062410

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