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：椭球模型边缘特征值的Tracy-Widom分布
翻译后的摘要：
本文研究了自变量服从椭圆分布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 i.i.d. 的 $p$ 维观测 $\mathbf{y}_i=\xi_iT\mathbf{u}_i,\;i=1,\dots,n$。这里 $\mathbf{u}_i$ 在单位球面上分布，$\xi_i \sim \xi$ 独立于 $\mathbf{u}_i$，且 $T^*T=\Sigma$ 是某个确定矩阵。在 $\Sigma$ 满足某些温和的正则性假设，$\xi^2$ 有有界支撑并且在其边缘处有适当的行为，使得 $Q$ 的极限谱分布（LSD）在其谱边缘附近具有平方衰减特性时，我们证明了当 $p$ 和 $n$ 逐渐增大时，$Q$ 的最大特征值符合 Tracy-Widom 定律。
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