AI 中文总结
研究可交换无条件随机向量\(\mathbf X\)的张量幂样本协方差矩阵，在适当矩条件下，证明其经验谱分布弱收敛到马尔琴科 - 帕斯特尔定律，扩展了坐标需独立的先前结果，适用于多种新随机向量。
AI 中文摘要
给定一个各向同性、可交换且无条件的随机向量\(\mathbf X\)，我们考虑由\(\mathbf X\)的几个张量模型（如张量幂\(\mathbf{X}^{\otimes d}\)）的独立同分布副本构建的样本协方差矩阵。在对\(\mathbf X\)的适当矩条件下，我们证明几乎必然地，经验谱分布弱收敛到马尔琴科 - 帕斯特尔定律。这扩展了先前要求\(\mathbf X\)的坐标独立的结果，且适用于许多新的感兴趣的随机向量\(\mathbf X\)。
英文摘要
Given an isotropic, exchangeable, and unconditional random vector $\mathbf X$, we consider the sample covariance matrix constructed from i.i.d. copies of several tensor models of $\mathbf X$, such as the tensor power $\mathbf{X}^{\otimes d}$. Under appropriate moment conditions on $\mathbf X$, we show that almost surely, the empirical spectral distribution converges weakly to the Marchenko-Pastur law. This extends previous results which required the coordinates of $\mathbf X$ to be independent. As we demonstrate, our extension applies to many new random vectors $\mathbf X$ of interest.
Comments33 pages
