## The Annals of Statistics

### The Tracy–Widom law for the largest eigenvalue of F type matrices

#### Abstract

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 $p\rightarrow\infty$.

#### Article information

Source
Ann. Statist., Volume 44, Number 4 (2016), 1564-1592.

Dates
Revised: December 2015
First available in Project Euclid: 7 July 2016

https://projecteuclid.org/euclid.aos/1467894708

Digital Object Identifier
doi:10.1214/15-AOS1427

Mathematical Reviews number (MathSciNet)
MR3519933

Zentralblatt MATH identifier
1378.60023

#### Citation

Han, Xiao; Pan, Guangming; Zhang, Bo. The Tracy–Widom law for the largest eigenvalue of F type matrices. Ann. Statist. 44 (2016), no. 4, 1564--1592. doi:10.1214/15-AOS1427. https://projecteuclid.org/euclid.aos/1467894708

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