UC Davis

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.