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December 2016 Tracy–Widom distribution for the largest eigenvalue of real sample covariance matrices with general population
Ji Oon Lee, Kevin Schnelli
Ann. Appl. Probab. 26(6): 3786-3839 (December 2016). DOI: 10.1214/16-AAP1193

Abstract

We consider sample covariance matrices of the form $\mathcal{Q}=(\Sigma^{1/2}X)(\Sigma^{1/2}X)^{*}$, where the sample $X$ is an $M\times N$ random matrix whose entries are real independent random variables with variance $1/N$ and where $\Sigma$ is an $M\times M$ positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of $\mathcal{Q}$ when both $M$ and $N$ tend to infinity with $N/M\to d\in(0,\infty)$. For a large class of populations $\Sigma$ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of $\mathcal{Q}$ is given by the type-1 Tracy–Widom distribution under the additional assumptions that (1) either the entries of $X$ are i.i.d. Gaussians or (2) that $\Sigma$ is diagonal and that the entries of $X$ have a sub-exponential decay.

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Ji Oon Lee. Kevin Schnelli. "Tracy–Widom distribution for the largest eigenvalue of real sample covariance matrices with general population." Ann. Appl. Probab. 26 (6) 3786 - 3839, December 2016. https://doi.org/10.1214/16-AAP1193

Information

Received: 1 June 2015; Revised: 1 January 2016; Published: December 2016
First available in Project Euclid: 15 December 2016

zbMATH: 1384.60026
MathSciNet: MR3582818
Digital Object Identifier: 10.1214/16-AAP1193

Subjects:
Primary: 15B52 , 60B20 , 62H10

Keywords: edge universality , Sample covariance matrix , Tracy–Widom distribution

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.26 • No. 6 • December 2016
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