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

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].