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

Xiucai Ding, Fan Yang

Published 2016-07-23Version 1

In this paper, we prove a necessary and sufficient condition for the edge universality of covariance matrices, which is conjectured by Pillai and Yin. Consider the sample covariance matrices of the form $XX^{*}$, where $X$ is an $M \times N$ rectangular matrix with i.i.d entries $X_{ij}=x_{ij}$ satisfying $\mathbb{E} x_{ij}=0$ and $ \mathbb{E} |x_{ij}|^2 ={N}^{-1}$. Under the assumption $\lim_{N \to \infty} {N}/{M}=d \in (0, \infty)$, we prove that the Tracy-Widom law holds at the soft edge (i.e., for the largest eigenvalues) if and only if $\lim_{s \rightarrow \infty}s^4 \mathbb{P}(\vert \sqrt{N} x_{ij} \vert \geq s)=0$. This condition was first proposed for Wigner matrices by Lee and Yin.