Tracy-Widom distribution for the edge eigenvalues of elliptical model

In this paper, we study the largest eigenvalues of sample covariance matrices with elliptically distributed data. We consider the sample covariance matrix $Q=Y{Y}^{\ast },$ where the data matrix $Y\in {\mathbb{R}}^{p\times n}$ contains i.i.d. $p$-dimensional observations ${\mathbf{y}}_i={\xi }_{i}T{\mathbf{u}}_i,i=1,\dots ,n\mathrm{.}$ Here ${\mathbf{u}}_i$ is distributed on the unit sphere, ${\xi }_i\sim \xi$ is independent of ${\mathbf{u}}_i$ and $T^{\ast }T=\mathrm{\Sigma }$ is some deterministic matrix. Under some mild regularity assumptions of $\mathrm{\Sigma },$ assuming ${\xi }^2$ has bounded support and certain proper behavior near its edge so that the limiting spectral distribution (LSD) of $Q$ has a square decay behavior 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.