Open Access
August 2016 The Tracy–Widom law for the largest eigenvalue of F type matrices
Xiao Han, Guangming Pan, Bo Zhang
Ann. Statist. 44(4): 1564-1592 (August 2016). DOI: 10.1214/15-AOS1427


Let ${\mathbf{A}}_{p}=\frac{{\mathbf{Y}}{\mathbf{Y}}^{*}}{m}$ and ${\mathbf{B}}_{p}=\frac{{\mathbf{X}}{\mathbf{X}}^{*}}{n}$ be two independent random matrices where ${\mathbf{X}}=(X_{ij})_{p\times n}$ and ${\mathbf{Y}}=(Y_{ij})_{p\times m}$ respectively consist of real (or complex) independent random variables with $\mathbb{E}X_{ij}=\mathbb{E}Y_{ij}=0$, $\mathbb{E}|X_{ij}|^{2}=\mathbb{E}|Y_{ij}|^{2}=1$. Denote by $\lambda_{1}$ the largest root of the determinantal equation $\det(\lambda{\mathbf{A}}_{p}-{\mathbf{B}}_{p})=0$. We establish the Tracy–Widom type universality for $\lambda_{1}$ under some moment conditions on $X_{ij}$ and $Y_{ij}$ when $p/m$ and $p/n$ approach positive constants as $p\rightarrow\infty$.


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Xiao Han. Guangming Pan. Bo Zhang. "The Tracy–Widom law for the largest eigenvalue of F type matrices." Ann. Statist. 44 (4) 1564 - 1592, August 2016.


Received: 1 June 2015; Revised: 1 December 2015; Published: August 2016
First available in Project Euclid: 7 July 2016

zbMATH: 1378.60023
MathSciNet: MR3519933
Digital Object Identifier: 10.1214/15-AOS1427

Primary: 34K25 , 60B20
Secondary: 60F05 , 62H10

Keywords: F matrix , Largest eigenvalue , Sample covariance matrix , Tracy–Widom distribution

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 4 • August 2016
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