Paper
Functional CLT for general sample covariance matrices
Authors
Jian Cui, Zhijun Liu, Jiang Hu, Zhidong Bai
Abstract
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 $κ>0$.
Metadata
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"raw_xml": "<entry>\n <id>http://arxiv.org/abs/2603.12780v1</id>\n <title>Functional CLT for general sample covariance matrices</title>\n <updated>2026-03-13T08:34:04Z</updated>\n <link href='https://arxiv.org/abs/2603.12780v1' rel='alternate' type='text/html'/>\n <link href='https://arxiv.org/pdf/2603.12780v1' rel='related' title='pdf' type='application/pdf'/>\n <summary>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 $κ>0$.</summary>\n <category scheme='http://arxiv.org/schemas/atom' term='math.ST'/>\n <category scheme='http://arxiv.org/schemas/atom' term='math.PR'/>\n <published>2026-03-13T08:34:04Z</published>\n <arxiv:primary_category term='math.ST'/>\n <author>\n <name>Jian Cui</name>\n </author>\n <author>\n <name>Zhijun Liu</name>\n </author>\n <author>\n <name>Jiang Hu</name>\n </author>\n <author>\n <name>Zhidong Bai</name>\n </author>\n </entry>"
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