The Tracy-Widom law for the Largest Eigenvalue of F Type Matrix

Abstract

Let ${\mathbb{A}}_p=\frac{\mathbb{Y}{\mathbb{Y}}^{\ast }}{m}$ and ${\mathbb{B}}_p=\frac{\mathbb{X}{\mathbb{X}}^{\ast }}{n}$ be two independent random matrices where $\mathbb{X}=(X_{ij}{)}_{p\times n}$ and $\mathbb{Y}=(Y_{ij}{)}_{p\times m}$ 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 $\det (\lambda {\mathbb{A}}_p-{\mathbb{B}}_p)=0$. 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 $p\to \infty$.

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References in corpus (1)

• Local semicircle law and complete delocalization for Wigner random matrices

Cited by in corpus (1)

• A unified matrix model including both CCA and F matrices in multivariate analysis: the largest eigenvalue and its applications