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 $\mathbf{y}_i=\xi_iT\mathbf{u}_i,\;i=1,\dots,n.$ Here $\mathbf{u}_i$ is distributed on the unit sphere, $\xi_i \sim \xi$ is independent of $\mathbf{u}_i$ and $T^*T=\Sigma$ is some deterministic matrix. Under some mild regularity assumptions of $\Sigma,$ 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.

翻译：本文研究了具有椭圆分布数据的样本协方差矩阵的最大特征值。我们考虑样本协方差矩阵$Q=YY^*,$ 其中数据矩阵$Y \in \mathbb{R}^{p \times n}$包含独立同分布的$p$维观测值$\mathbf{y}_i=\xi_iT\mathbf{u}_i,\;i=1,\dots,n.$ 这里$\mathbf{u}_i$分布在单位球面上，$\xi_i \sim \xi$独立于$\mathbf{u}_i$，且$T^*T=\Sigma$为某个确定性矩阵。在$\Sigma$的某些温和正则性假设下，假设$\xi^2$具有有界支撑且在其边缘附近具有适当的性质，使得$Q$的极限谱分布(LSD)在谱边缘附近呈现平方衰减行为，我们证明当$p$与$n$可比拟时，$Q$的最大特征值服从Tracy-Widom律。