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2019 Edge universality of separable covariance matrices
Fan Yang
Electron. J. Probab. 24: 1-57 (2019). DOI: 10.1214/19-EJP381


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. $\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 [6, 39] (assuming high moments and diagonal $A$), [37] (assuming high moments) and [14] (assuming diagonal $A$).


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Fan Yang. "Edge universality of separable covariance matrices." Electron. J. Probab. 24 1 - 57, 2019.


Received: 12 May 2019; Accepted: 24 October 2019; Published: 2019
First available in Project Euclid: 6 November 2019

zbMATH: 07142917
MathSciNet: MR4029426
Digital Object Identifier: 10.1214/19-EJP381

Primary: 15B52
Secondary: 62E20, 62H99


Vol.24 • 2019
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