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In this note, we consider a sample covariance matrix of the form \(\displaystyle M_{n}=\sum _{\alpha =1}^m \tau _\alpha {\mathbf {y}}_{\alpha }^{(1)} \otimes {\mathbf {y}}_{\alpha }^{(2)}({\mathbf {y}}_{\alpha }^{(1)} \otimes {\mathbf {y}}_{\alpha }^{(2)})^T, \) 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-Kolodziejczyk and Lytova, J Math Phys Anal Geom 19(2):1–22, 2023; Lytova, J Theoret Probab 31(2):1024–1057, 2018.

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A Note on the Fluctuations of the Resolvent Traces of a Tensor Model of Sample Covariance Matrices

  • 会议论文

  • 发表日期：

    2026年

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  • Alicja Dembczak-Kolodziejczyk

摘要

In this note, we consider a sample covariance matrix of the form \(\displaystyle M_{n}=\sum _{\alpha =1}^m \tau _\alpha {\mathbf {y}}_{\alpha }^{(1)} \otimes {\mathbf {y}}_{\alpha }^{(2)}({\mathbf {y}}_{\alpha }^{(1)} \otimes {\mathbf {y}}_{\alpha }^{(2)})^T, \) 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-Kolodziejczyk and Lytova, J Math Phys Anal Geom 19(2):1–22, 2023; Lytova, J Theoret Probab 31(2):1024–1057, 2018.