Tracy–Widom distribution for the edge eigenvalues of elliptical model

Abstract In this paper, we study the largest eigenvalues of sample covariance matrices with elliptically distributed data. We consider the sample covariance matrix $Q=YY^{*},$ where the data matrix $Y \in \mathbb{R}^{p \times n}$ contains i.i.d. $p$-dimensional observations $\textbf{y}_{i}=\xi _{i}T\textbf{u}_{i},\;i=1,\dots ,n.$ Here $\textbf{u}_{i}$ is distributed on the unit sphere, $\xi _{i} \sim \xi $ is some random variable that is independent of $\textbf{u}_{i}$ and $T^{*}T=\varSigma $ is some deterministic positive definite matrix. Under some mild regularity assumptions on $\varSigma ,$ assuming $\xi ^{2}$ has bounded support and certain decay behaviour near its edge so that the limiting spectral distribution of $Q$ has a square root decay behaviour near the spectral edge, we prove that the Tracy–Widom law holds for the largest eigenvalues of $Q$ when $p$ and $n$ are comparably large. Based on our results, we further construct some useful statistics to detect the signals when they are corrupted by high dimensional elliptically distributed noise.