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

By Xiucai Ding, Fan Yang. The Annals of Applied Probability, 2016

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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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3a2e4bacaab0a95f3f0dd8507c5960767e3b516a

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In Discrete Mathematics and Combinatorics

  1. Xiucai Ding
  2. Fan Yang

In Mathematical Physics

  1. Fan Yang
  2. Xiucai Ding

In Statistics and Probability

  1. Xiucai Ding
  2. Fan Yang

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