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June 2018 A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices
Xiucai Ding, Fan Yang
Ann. Appl. Probab. 28(3): 1679-1738 (June 2018). DOI: 10.1214/17-AAP1341

Abstract

In this paper, we prove a necessary and sufficient condition for the edge universality of sample covariance matrices with general population. We consider sample covariance matrices of the form $\mathcal{Q}=TX(TX)^{*}$, where $X$ is an $M_{2}\times N$ random matrix with $X_{ij}=N^{-1/2}q_{ij}$ such that $q_{ij}$ are i.i.d. random variables with zero mean and unit variance, and $T$ is an $M_{1}\times M_{2}$ deterministic matrix such that $T^{*}T$ is diagonal. We study the asymptotic behavior of the largest eigenvalues of $\mathcal{Q}$ when $M:=\min\{M_{1},M_{2}\}$ and $N$ tend to infinity with $\lim_{N\to\infty}{N}/{M}=d\in(0,\infty)$. We prove that the Tracy–Widom law holds for the largest eigenvalue of $\mathcal{Q}$ if and only if $\lim_{s\rightarrow\infty}s^{4}\mathbb{P}(\vert q_{ij}\vert\geq s)=0$ under mild assumptions of $T$. The necessity and sufficiency of this condition for the edge universality was first proved for Wigner matrices by Lee and Yin [Duke Math. J. 163 (2014) 117–173].

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Xiucai Ding. Fan Yang. "A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices." Ann. Appl. Probab. 28 (3) 1679 - 1738, June 2018. https://doi.org/10.1214/17-AAP1341

Information

Received: 1 March 2017; Revised: 1 August 2017; Published: June 2018
First available in Project Euclid: 1 June 2018

zbMATH: 06919736
MathSciNet: MR3809475
Digital Object Identifier: 10.1214/17-AAP1341

Subjects:
Primary: 15B52 , 82B44

Keywords: edge universality , Marchenko–Pastur law , Sample covariance matrices , Tracy–Widom distribution

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 3 • June 2018
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