加核可分离协方差矩阵的局部律与离群特征值
Local Law and Outlier Eigenvalues of Spiked Separable Covariance Matrices
- Qiuzhen College, Tsinghua University（清华大学邱氏书院）
- Department of Probability and Statistics, School of Mathematical Sciences, Peking University（北京大学数学科学学院概率统计系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对加核可分离协方差矩阵的预解式证明局部律，无需先前必要的三阶矩为零假设，并将其应用于推导加核可分离协方差矩阵离群特征值的渐近分布，扩展了相关已有结果。
AI中文摘要：
我们针对形如$\boldsymbol{\text{Q}}=A^{1/2}XBX^*A^{1/2}$的可分离协方差矩阵的预解式证明了局部律，其中$X=(x_{ij})$是$p\times n$随机矩阵，其元素$x_{ij}$为均值0、方差$n^{-1}$的独立同分布随机变量，$A,B$为确定性非负定对称（或埃尔米特）矩阵。遵循arXiv:1611.05364提出的方法，我们首先建立$\boldsymbol{\text{Q}}$预解式的自洽方程，并利用它证明了无需技术假设$\boldsymbol{\text{E}}[x_{ij}^{3}]=0$的最优局部律，该假设是arXiv:1809.04572中局部律先前推导的必要条件。作为我们局部律的应用，我们计算了加核可分离协方差矩阵离群特征值的渐近分布，扩展了arXiv:2008.11903中的相应结果。
英文摘要：
We prove local laws for the resolvents of separable covariance matrices of the form $\mathcal Q=A^{1/2}XBX^*A^{1/2}$, where $X=(x_{ij})$ is a $p\times n$ random matrix whose entries $x_{ij}$ are i.i.d.~random variables with mean 0 and variance $n^{-1}$, and $A,B$ are deterministic non-negative definite symmetric (or Hermitian) matrices. Following the method developed in arXiv:1611.05364, we first establish a self-consistent equation for the resolvent of $\mathcal Q$ and use it to prove optimal local laws without the technical assumption $\mathbb{E}[x_{ij}^{3}]=0$, which was essential in the previous derivation of the local laws in arXiv:1809.04572. As an application of our local law, we compute the asymptotic distribution of the outlier eigenvalues for spiked separable covariance matrices, extending the corresponding result in arXiv:2008.11903.
