The Annals of Statistics

Edge universality of correlation matrices

Natesh S. Pillai and Jun Yin

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Abstract

Let $\widetilde{X}_{M\times N}$ be a rectangular data matrix with independent real-valued entries $[\widetilde{x}_{ij}]$ satisfying $\mathbb{E} \widetilde{x}_{ij}=0$ and $\mathbb{E} \widetilde{x}^{2}_{ij}=\frac{1}{M}$, $N,M\to\infty$. These entries have a subexponential decay at the tails. We will be working in the regime $N/M=d_{N}$, $\lim_{N\to\infty}d_{N}\neq0,1,\infty$. In this paper we prove the edge universality of correlation matrices ${X}^{\dagger}X$, where the rectangular matrix $X$ (called the standardized matrix) is obtained by normalizing each column of the data matrix $\widetilde{X}$ by its Euclidean norm. Our main result states that asymptotically the $k$-point ($k\geq1$) correlation functions of the extreme eigenvalues (at both edges of the spectrum) of the correlation matrix ${X}^{\dagger}X$ converge to those of the Gaussian correlation matrix, that is, Tracy–Widom law, and, thus, in particular, the largest and the smallest eigenvalues of ${X}^{\dagger}X$ after appropriate centering and rescaling converge to the Tracy–Widom distribution. The asymptotic distribution of extreme eigenvalues of the Gaussian correlation matrix has been worked out only recently. As a corollary of the main result in this paper, we also obtain that the extreme eigenvalues of Gaussian correlation matrices are asymptotically distributed according to the Tracy–Widom law. The proof is based on the comparison of Green functions, but the key obstacle to be surmounted is the strong dependence of the entries of the correlation matrix. We achieve this via a novel argument which involves comparing the moments of product of the entries of the standardized data matrix to those of the raw data matrix. Our proof strategy may be extended for proving the edge universality of other random matrix ensembles with dependent entries and hence is of independent interest.

Article information

Source
Ann. Statist., Volume 40, Number 3 (2012), 1737-1763.

Dates
First available in Project Euclid: 2 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.aos/1349196390

Digital Object Identifier
doi:10.1214/12-AOS1022

Mathematical Reviews number (MathSciNet)
MR3015042

Zentralblatt MATH identifier
1260.15051

Subjects
Primary: 15B52: Random matrices 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Covariance matrix correlation matrix Marcenko–Pastur law universality

Citation

Pillai, Natesh S.; Yin, Jun. Edge universality of correlation matrices. Ann. Statist. 40 (2012), no. 3, 1737--1763. doi:10.1214/12-AOS1022. https://projecteuclid.org/euclid.aos/1349196390


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