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

Xiucai Ding University of Toronto Fan Yang University of Wisconsin-Madison

Probability mathscidoc:1805.28001

Annals of Applied Probability, 2018
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 [34].
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@inproceedings{xiucai2018a,
  title={A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices},
  author={Xiucai Ding, and Fan Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180507141351094707083},
  booktitle={Annals of Applied Probability},
  year={2018},
}
Xiucai Ding, and Fan Yang. A necessary and sufficient condition for edge universality at the largest singular values of covariance matrices. 2018. In Annals of Applied Probability. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180507141351094707083.
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