Edge universality of separable covariance matrices

TL;DR

This paper proves that the largest eigenvalues of a broad class of separable covariance matrices follow the same distribution as in the Gaussian case, under mild conditions, extending universality results to correlated and heavy-tailed data.

Contribution

It establishes the strongest known edge universality results for separable covariance matrices with correlated data and heavy tails, relaxing previous moment and structure assumptions.

Findings

Largest eigenvalues follow Gaussian ensemble distribution

Edge universality holds under minimal moment conditions

Results extend to matrices with correlated data and heavy tails

Abstract

In this paper, we prove the edge universality of largest eigenvalues for separable covariance matrices of the form $\mathcal Q :=A^{1/2}XBX^*A^{1/2}$. Here $X=(x_{ij})$ is an $n\times N$ random matrix with $x_{ij}=N^{-1/2}q_{ij}$, where $q_{ij}$ are $i.i.d.$ random variables with zero mean and unit variance, and $A$ and $B$ are respectively $n \times n$ and $N\times N$ deterministic non-negative definite symmetric (or Hermitian) matrices. We consider the high-dimensional case, i.e. ${n}/{N}\to d \in (0, \infty)$ as $N\to \infty$. Assuming $\mathbb E q_{ij}^3=0$ and some mild conditions on $A$ and $B$, we prove that the limiting distribution of the largest eigenvalue of $\mathcal Q$ coincide with that of the corresponding Gaussian ensemble (i.e. the $\mathcal Q$ with $X$ being an $i.i.d.$ Gaussian matrix) as long as we have $\lim_{s \rightarrow \infty}s^4 \mathbb{P}(\vert q_{ij} \vert \geq s)=0$, which is a sharp moment condition for edge universality. If we take $B=I$, then $\mathcal Q$ becomes the normal sample covariance matrix and the edge universality holds true without the vanishing third moment condition. So far, this is the strongest edge universality result for sample covariance matrices with correlated data (i.e. non-diagonal $A$) and heavy tails, which improves the previous results in \cite{BPZ1,LS} (assuming high moments and diagonal $A$), \cite{Anisotropic} (assuming high moments) and \cite{DY} (assuming diagonal $A$).