Functional CLT for general sample covariance matrices
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866912964672487424 |
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| author | Cui, Jian Liu, Zhijun Hu, Jiang Bai, Zhidong |
| author_facet | Cui, Jian Liu, Zhijun Hu, Jiang Bai, Zhidong |
| contents | This paper studies the central limit theorems (CLTs) for linear spectral statistics (LSSs) of general sample covariance matrices, when the test functions belong to $C^3$, the class of functions with continuous third order derivatives. We consider matrices of the form $B_n=(1/n)T_p^{1/2}X_nX_n^{*}T_p^{1/2},$ where $X_n= (x_{i j} ) $ is a $p \times n$ matrix whose entries are independent and identically distributed (i.i.d.) real or complex random variables, and $T_p$ is a $p\times p$ nonrandom Hermitian nonnegative definite matrix with its spectral norm uniformly bounded in $p$. By using Bernstein polynomial approximation, we show that, under $\mathbb{E}|x_{ij}|^{8}<\infty$, the centered LSSs of $B_n$ have Gaussian limits. Under the stronger $\mathbb{E}|x_{ij}|^{10}<\infty$, we further establish convergence rates $O(n^{-1/2+κ})$ in Kolmogorov--Smirnov $O(n^{-1/2+κ})$, for any fixed $κ>0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_12780 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Functional CLT for general sample covariance matrices Cui, Jian Liu, Zhijun Hu, Jiang Bai, Zhidong Statistics Theory Probability This paper studies the central limit theorems (CLTs) for linear spectral statistics (LSSs) of general sample covariance matrices, when the test functions belong to $C^3$, the class of functions with continuous third order derivatives. We consider matrices of the form $B_n=(1/n)T_p^{1/2}X_nX_n^{*}T_p^{1/2},$ where $X_n= (x_{i j} ) $ is a $p \times n$ matrix whose entries are independent and identically distributed (i.i.d.) real or complex random variables, and $T_p$ is a $p\times p$ nonrandom Hermitian nonnegative definite matrix with its spectral norm uniformly bounded in $p$. By using Bernstein polynomial approximation, we show that, under $\mathbb{E}|x_{ij}|^{8}<\infty$, the centered LSSs of $B_n$ have Gaussian limits. Under the stronger $\mathbb{E}|x_{ij}|^{10}<\infty$, we further establish convergence rates $O(n^{-1/2+κ})$ in Kolmogorov--Smirnov $O(n^{-1/2+κ})$, for any fixed $κ>0$. |
| title | Functional CLT for general sample covariance matrices |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2603.12780 |