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

TL;DR

This paper establishes a precise condition under which the largest eigenvalues of certain covariance matrices follow the Tracy-Widom distribution, linking edge universality to tail decay of matrix entries.

Contribution

It provides a necessary and sufficient condition for edge universality of covariance matrices with general population, extending previous results to a broader class of matrices.

Findings

Tracy-Widom law applies under specific tail decay conditions.

The condition is both necessary and sufficient for edge universality.

Results generalize prior work on Wigner matrices to 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 the sample $X$ is an $M_2\times N$ random matrix with $i.i.d.$ entries with mean zero and variance $N^{-1}$, and $T$ is an $M_1 \times M_2$ deterministic matrix satisfying $T^* T$ is diagonal. We study the asymptotic behavior of the largest eigenvalues of $\mathcal Q$ when $M:=\min\{M_1,M_2\}$ and $N$ tends to infinity with $\lim_{N \to \infty} {N}/{M}=d \in (0, \infty)$. Under mild assumptions of $T$, we prove that the Tracy-Widom law holds for the largest eigenvalue of $\mathcal Q$ if and only if $\lim_{s \rightarrow \infty}s^4 \mathbb{P}(| \sqrt{N} x_{ij}| \geq s)=0$. This condition was first proposed for Wigner matrices by Lee and Yin.