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

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1.	Title	Title of document	Tracy-Widom law for the extreme eigenvalues of sample correlation matrices
2.	Creator	Author's name, affiliation, country	Zhigang Bao; Zhejiang University; China
2.	Creator	Author's name, affiliation, country	Guangming Pan; Nanyang Technological University; Singapore
2.	Creator	Author's name, affiliation, country	Wang Zhou; National University of Singapore; Singapore
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	extreme eigenvalues; sample correlation matrices; sample covariance matrices; Stieltjes transform; Tracy-Widom law
3.	Subject	Subject classification	15B52; 62H25; 62H10
4.	Description	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 $0<p<n$ 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 $r_{ij}=(x_{ij}-\bar x_i)/\sqrt{\sum_{j=1}^n(x_{ij}-\bar x_i)^2}$, $\bar x_i=n^{-1}\sum_{j=1}^nx_{ij}$.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2012-10-04
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/1962
10.	Identifier	Digital Object Identifier	10.1214/EJP.v17-1962
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 17
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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