A note on the fluctuations of the resolvent traces of a tensor model of sample covariance matrices
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arXiv
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| Format: | Preprint |
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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 |