On the CLT for Linear Eigenvalue Statistics of a Tensor Model of Sample Covariance Matrices
In [18], there was proved the CLT for linear eigenvalue statistics ${Tr} \varphi(M_n)$ of sample covariance matrices of the form $M_{n}=\sum_{\alpha=1}^m {\bf y}_{\alpha}^{(1)} \otimes {\bf y}_{\alpha}^{(2)}({\bf y}_{\alpha}^{(1)} \otimes {\bf y}_{\alpha}^{(2)})^T$, where $({\bf y}_{\alpha}^{(1)},\,...
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| Date: | 2023 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України
2023
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| Subjects: | |
| Online Access: | https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1012 |
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| Journal Title: | Journal of Mathematical Physics, Analysis, Geometry |
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Journal of Mathematical Physics, Analysis, Geometry| Summary: | In [18], there was proved the CLT for linear eigenvalue statistics ${Tr} \varphi(M_n)$ of sample covariance matrices of the form $M_{n}=\sum_{\alpha=1}^m {\bf y}_{\alpha}^{(1)} \otimes {\bf y}_{\alpha}^{(2)}({\bf y}_{\alpha}^{(1)} \otimes {\bf y}_{\alpha}^{(2)})^T$, where $({\bf y}_{\alpha}^{(1)},\, {\bf y}_{\alpha}^{(2)})_{\alpha}$ are iid copies of ${\bf y}\in \mathbb{R}^n$ satisfying ${\bf E} {\bf y}{\bf y}^T=n^{-1} I_n$, ${\bf E} {\bf y}^2_i{\bf y}^2_j=(1+\delta_{ij}d)n^{-2}+a(1+\delta_{ij}d_1)n^{-3}+O(n^{-4})$ for some $a, d,d_1\in \mathbb{R}$. It was shown that given a smooth enough test function $\varphi$, ${\bf Var} {Tr} \varphi(M_n)=O(n)$ as $m, n\to\infty $, $m/n^2\to c>0$, and $({Tr} \varphi(M_n)-{\bf E} {Tr} \varphi(M_n))/\sqrt{n}$ converges in distribution to a Gaussian mean zero random variable with variance $V[\varphi]$ proportional to $a+d$. It was noticed that if ${\bf y}$ is uniformly distributed on the unit sphere then $a+d=0$ and $V[\varphi]$ vanishes. In this note we show that in this case ${\bf Var} {Tr}(M_n-zI_n)^{-1}=O(1)$, so that the CLT should be valid for linear eigenvalue statistics themselves without a normalising factor in front (in contrast to the Gaussian case.)
Mathematical Subject Classification 2020: 60B12, 60B20, 60F05, 47A75 |
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