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:对椭圆模型的边界特征值的Tracy-Widom分布
翻译后的摘要:
本文研究身材呈椭圆形的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 $Q=YY^*,$ 这里数据矩阵 $Y \in \mathbb{R}^{p \times n}$ 包含 i.i.d. $p$ 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。在对 $\Sigma$ 进行一些温和的正则化假设的情况下,$\xi^2$有界支持,且在其边缘附近具有某种适当的行为,使得当 $p$ 和 $n$ 都相对较大时,$Q$ 的极限谱分布 (LSD) 在谱边缘附近具有平方衰减行为,我们证明了 $Q$ 的最大特征值的Tracy-Widom定律成立。