关于样本协方差矩阵张量模型解析痕量波动的说明

Alicja Dembczak-Kołodziejczyk
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引用次数: 0

摘要

在本说明中我们考虑的样本协方差矩阵的形式为$$M_{n}=\sum_{\alpha=1}^m \tau_\alpha {\mathbf{y}}_\{alpha}^{(1)}\otimes{\mathbf{y}}_{\alpha}^{(2)}({\mathbf{y}}_{\alpha}^{(1)} \otimes{\mathbf{y}}_{\alpha}^{(2)})^T,$$ 其中 $(\mathbf{y}_{\alpha}^{(1)},\,{\mathbf{y}}_{\alpha}^{(2)})_{\alpha}$ 是均匀分布在单位球面 $S^{n-1}$ 上的独立向量,并且 $\tau_\alpha \ in \mathbb{R}_+ $.我们证明,当 $m, n 到 \infty$, $m/n^2\to c>0$ 时,解析子的集中迹线,$\mathrm{Tr}(M_n-zI_n)^{-1}-\mathbf{E}\mathrm{Tr}(M_n-zI_n)^{-1}$、$\Im z\ge\eta_0>0$, 在分布上收敛于具有零均值和一定协方差矩阵的二维高斯随机变量。这项工作是 Dembczak-Ko{\l}odziejczyk 和 Lytova (2023) 以及 Lytova (2018) 的继续。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A note on the fluctuations of the resolvent traces of a tensor model of sample covariance matrices
In this note, we consider a sample covariance matrix of the form $$M_{n}=\sum_{\alpha=1}^m \tau_\alpha {\mathbf{y}}_{\alpha}^{(1)} \otimes {\mathbf{y}}_{\alpha}^{(2)}({\mathbf{y}}_{\alpha}^{(1)} \otimes {\mathbf{y}}_{\alpha}^{(2)})^T,$$ where $(\mathbf{y}_{\alpha}^{(1)},\, {\mathbf{y}}_{\alpha}^{(2)})_{\alpha}$ are independent vectors uniformly distributed on the unit sphere $S^{n-1}$ and $\tau_\alpha \in \mathbb{R}_+ $. We show that as $m, n \to \infty$, $m/n^2\to c>0$, the centralized traces of the resolvents, $\mathrm{Tr}(M_n-zI_n)^{-1}-\mathbf{E}\mathrm{Tr}(M_n-zI_n)^{-1}$, $\Im z\ge \eta_0>0$, converge in distribution to a two-dimensional Gaussian random variable with zero mean and a certain covariance matrix. This work is a continuation of Dembczak-Ko{\l}odziejczyk and Lytova (2023), and Lytova (2018).
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