The Annals of Applied Probability

A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices

Xiucai Ding and Fan Yang

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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].

Article information

Source
Ann. Appl. Probab., Volume 28, Number 3 (2018), 1679-1738.

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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1527840030

Digital Object Identifier
doi:10.1214/17-AAP1341

Mathematical Reviews number (MathSciNet)
MR3809475

Zentralblatt MATH identifier
06919736

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

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

Citation

Ding, Xiucai; Yang, Fan. A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices. Ann. Appl. Probab. 28 (2018), no. 3, 1679--1738. doi:10.1214/17-AAP1341. https://projecteuclid.org/euclid.aoap/1527840030


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