A note on the fluctuations of the resolvent traces of a tensor model of sample covariance matrices

Fuente: arXiv
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Autor principal: Dembczak-Kołodziejczyk, Alicja
Formato: Preprint
Publicado: 2024
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author Dembczak-Kołodziejczyk, Alicja
author_facet Dembczak-Kołodziejczyk, Alicja
contents In this note, we consider a sample covariance matrix of the form $$M_{n}=\sum_{α=1}^m τ_α{\mathbf{y}}_α^{(1)} \otimes {\mathbf{y}}_α^{(2)}({\mathbf{y}}_α^{(1)} \otimes {\mathbf{y}}_α^{(2)})^T,$$ where $(\mathbf{y}_α^{(1)},\, {\mathbf{y}}_α^{(2)})_α$ are independent vectors uniformly distributed on the unit sphere $S^{n-1}$ and $τ_α\in \mathbb{R}_+ $. We show that as $m, n \to \infty$, $m/n^2\to c>0$, the centralized traces of the resolvents, $\mathrm{Tr}(M_n-zI_n)^{-1}-\mathbf{E}\mathrm{Tr}(M_n-zI_n)^{-1}$, $\Im z\ge η_0>0$, 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-Kołodziejczyk and Lytova (2023), and Lytova (2018).
format Preprint
id arxiv_https___arxiv_org_abs_2409_06007
institution arXiv
publishDate 2024
record_format arxiv
spellingShingle A note on the fluctuations of the resolvent traces of a tensor model of sample covariance matrices
Dembczak-Kołodziejczyk, Alicja
Probability
In this note, we consider a sample covariance matrix of the form $$M_{n}=\sum_{α=1}^m τ_α{\mathbf{y}}_α^{(1)} \otimes {\mathbf{y}}_α^{(2)}({\mathbf{y}}_α^{(1)} \otimes {\mathbf{y}}_α^{(2)})^T,$$ where $(\mathbf{y}_α^{(1)},\, {\mathbf{y}}_α^{(2)})_α$ are independent vectors uniformly distributed on the unit sphere $S^{n-1}$ and $τ_α\in \mathbb{R}_+ $. We show that as $m, n \to \infty$, $m/n^2\to c>0$, the centralized traces of the resolvents, $\mathrm{Tr}(M_n-zI_n)^{-1}-\mathbf{E}\mathrm{Tr}(M_n-zI_n)^{-1}$, $\Im z\ge η_0>0$, 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-Kołodziejczyk and Lytova (2023), and Lytova (2018).
title A note on the fluctuations of the resolvent traces of a tensor model of sample covariance matrices
topic Probability
url https://arxiv.org/abs/2409.06007