The Geometry of Spectral Fluctuations: On Near-Optimal Conditions for Universal Gaussian CLTs, with Statistical Applications
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918496018890752 |
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| author | Yin, Yanqing Zhou, Wang |
| author_facet | Yin, Yanqing Zhou, Wang |
| contents | We study linear spectral statistics of high dimensional sample covariance matrices in a regime where the empirical spectral distribution remains governed by the classical sample covariance law but the fluctuation theory is nonclassical. Our starting point is a decomposition of the covariance of centered quadratic forms into a universal Gaussian part and a model dependent fourth order correction. This leads to an abstract framework, termed GHOST, for universal Gaussian central limit theorems under structured fourth order effects. Under this framework, we prove a Gaussian central limit theorem for linear spectral statistics, with explicit mean and covariance corrections determined by a bilinear fourth order kernel. Boundary examples show that the conditions are close to necessary for a broad universal Gaussian closure. We then develop a blockwise mixed radial model that verifies the abstract assumptions and makes the correction explicit. The correction splits into an entrywise fourth moment component and a lockwise energy fluctuation component. The latter may change the fluctuation scale, leading to a phase transition at the level of fluctuations. As an application, we study sphericity testing. Under the spherical null, the general correction collapses to a single scalar parameter, yielding a feasible data driven correction of John's test. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11372 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Geometry of Spectral Fluctuations: On Near-Optimal Conditions for Universal Gaussian CLTs, with Statistical Applications Yin, Yanqing Zhou, Wang Statistics Theory 62H15, 62B20 We study linear spectral statistics of high dimensional sample covariance matrices in a regime where the empirical spectral distribution remains governed by the classical sample covariance law but the fluctuation theory is nonclassical. Our starting point is a decomposition of the covariance of centered quadratic forms into a universal Gaussian part and a model dependent fourth order correction. This leads to an abstract framework, termed GHOST, for universal Gaussian central limit theorems under structured fourth order effects. Under this framework, we prove a Gaussian central limit theorem for linear spectral statistics, with explicit mean and covariance corrections determined by a bilinear fourth order kernel. Boundary examples show that the conditions are close to necessary for a broad universal Gaussian closure. We then develop a blockwise mixed radial model that verifies the abstract assumptions and makes the correction explicit. The correction splits into an entrywise fourth moment component and a lockwise energy fluctuation component. The latter may change the fluctuation scale, leading to a phase transition at the level of fluctuations. As an application, we study sphericity testing. Under the spherical null, the general correction collapses to a single scalar parameter, yielding a feasible data driven correction of John's test. |
| title | The Geometry of Spectral Fluctuations: On Near-Optimal Conditions for Universal Gaussian CLTs, with Statistical Applications |
| topic | Statistics Theory 62H15, 62B20 |
| url | https://arxiv.org/abs/2605.11372 |