How to cite item

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

   
@article{EJP1962,
	author = {Zhigang Bao and Guangming Pan and Wang Zhou},
	title = {Tracy-Widom law for the extreme eigenvalues of sample correlation matrices},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {17},
	year = {2012},
	keywords = {extreme eigenvalues; sample correlation matrices; sample covariance matrices; Stieltjes transform; Tracy-Widom law},
	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 $r_{ij}=(x_{ij}-\bar  x_i)/\sqrt{\sum_{j=1}^n(x_{ij}-\bar x_i)^2}$, $\bar x_i=n^{-1}\sum_{j=1}^nx_{ij}$. 
},
	pages = {no. 88, 1-32},
	issn = {1083-6489},
	doi = {10.1214/EJP.v17-1962},    
        url = {http://ejp.ejpecp.org/article/view/1962}}