On the rate of convergence in the CLT for LSS of large-dimensional sample covariance matrices
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918045307371520 |
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| author | Cui, Jian Hu, Jiang Bai, Zhidong Hu, Guorong |
| author_facet | Cui, Jian Hu, Jiang Bai, Zhidong Hu, Guorong |
| contents | This paper investigates the rate of convergence for the central limit theorem of linear spectral statistic (LSS) associated with large-dimensional sample covariance matrices. We consider matrices of the form ${\mathbf B}_n=\frac{1}{n}{\mathbf T}_p^{1/2}{\mathbf X}_n{\mathbf X}_n^*{\mathbf T}_p^{1/2},$ where ${\mathbf X}_n= (x_{i j} ) $ is a $p \times n$ matrix whose entries are independent and identically distributed (i.i.d.) real or complex variables, and ${\mathbf T} _p$ is a $p\times p$ nonrandom Hermitian nonnegative definite matrix with its spectral norm uniformly bounded in $p$. Employing Stein's method, we establish that if the entries $x_{ij}$ satisfy $\mathbb{E}|x_{ij}|^{10}<\infty$ and the ratio of the dimension to sample size $p/n\to y>0$ as $n\to\infty$, then the convergence rate of the normalized LSS of ${\mathbf B}_n$ to the standard normal distribution, measured in the Kolmogorov-Smirnov distance, is $O(n^{-1/2+κ})$ for any fixed $κ>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_02880 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the rate of convergence in the CLT for LSS of large-dimensional sample covariance matrices Cui, Jian Hu, Jiang Bai, Zhidong Hu, Guorong Probability Statistics Theory This paper investigates the rate of convergence for the central limit theorem of linear spectral statistic (LSS) associated with large-dimensional sample covariance matrices. We consider matrices of the form ${\mathbf B}_n=\frac{1}{n}{\mathbf T}_p^{1/2}{\mathbf X}_n{\mathbf X}_n^*{\mathbf T}_p^{1/2},$ where ${\mathbf X}_n= (x_{i j} ) $ is a $p \times n$ matrix whose entries are independent and identically distributed (i.i.d.) real or complex variables, and ${\mathbf T} _p$ is a $p\times p$ nonrandom Hermitian nonnegative definite matrix with its spectral norm uniformly bounded in $p$. Employing Stein's method, we establish that if the entries $x_{ij}$ satisfy $\mathbb{E}|x_{ij}|^{10}<\infty$ and the ratio of the dimension to sample size $p/n\to y>0$ as $n\to\infty$, then the convergence rate of the normalized LSS of ${\mathbf B}_n$ to the standard normal distribution, measured in the Kolmogorov-Smirnov distance, is $O(n^{-1/2+κ})$ for any fixed $κ>0$. |
| title | On the rate of convergence in the CLT for LSS of large-dimensional sample covariance matrices |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2506.02880 |