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

Abstract

Let the sample correlation matrix be $W=Y{Y}^T$, where $Y=(y_{ij}{)}_{p,n}$ with $y_{ij}=x_{ij}/\sqrt{{\sum }_{j=1}^n{x}_{ij}^2}$. We assume $\{x_{ij}:1\le i\le p,1\le j\le n\}$ to be a collection of independent symmetric distributed random variables with sub-exponential tails. Moreover, for any $i$, we assume $x_{ij},1\le j\le n$ to be identically distributed. We assume $0<p<n$ and $p/n\to y$ with some $y\in (0,1)$ as $p,n\to \infty$. In this paper, we provide the Tracy-Widom law ($T{W}_1$) for both the largest and smallest eigenvalues of $W$. If $x_{ij}$ are i.i.d. standard normal, we can derive the $T{W}_1$ for both the largest and smallest eigenvalues of the matrix $\mathcal{R}=R{R}^T$, where $R=(r_{ij}{)}_{p,n}$ with $r_{ij}=(x_{ij}-{\bar{x}}_i)/\sqrt{{\sum }_{j=1}^n(x_{ij}-{\bar{x}}_i{)}^2}$, ${\bar{x}}_i=n^{-1}{\sum }_{j=1}^n{x}_{ij}$.

35 pages, a major revision

References in corpus (1)

• Universality of covariance matrices

Cited by in corpus (1)

• Edge universality of correlation matrices