Quantitative bounds for high-dimensional non-linear functionals of Gaussian processes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917239880417280 |
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| author | Basse-O'Connor, Andreas Kramer-Bang, David |
| author_facet | Basse-O'Connor, Andreas Kramer-Bang, David |
| contents | In this paper, we establish explicit quantitative Berry-Esseen bounds in the hyper-rectangle distance $d_R$, the convex distance $d_{\mathscr{C}}$ and the $1$-Wasserstein distance $d_W$ for high-dimensional, non-linear functionals of Gaussian processes, allowing for strong dependence between variables. Our main result demonstrates that, under a smoothness assumption, the convergence rate under $d_R$ is sub-polynomial in the dimension and polynomial under $d_{\mathscr{C}}$ and $d_W$. To the best of our knowledge, our results under $d_R$ provide the first explicit sub-polynomial bound for high-dimensional, non-linear functionals of Gaussian processes beyond the i.i.d. setting. Building on this, we derive explicit Berry-Esseen bounds under both $d_R$ and $d_{\mathscr{C}}$ for multiple statistical examples, such as the method of moments, empirical characteristic functions, empirical moment-generating functions, and functional limit theorems in high-dimensional settings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_17718 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative bounds for high-dimensional non-linear functionals of Gaussian processes Basse-O'Connor, Andreas Kramer-Bang, David Probability 60F05, 60G15, 60H07 In this paper, we establish explicit quantitative Berry-Esseen bounds in the hyper-rectangle distance $d_R$, the convex distance $d_{\mathscr{C}}$ and the $1$-Wasserstein distance $d_W$ for high-dimensional, non-linear functionals of Gaussian processes, allowing for strong dependence between variables. Our main result demonstrates that, under a smoothness assumption, the convergence rate under $d_R$ is sub-polynomial in the dimension and polynomial under $d_{\mathscr{C}}$ and $d_W$. To the best of our knowledge, our results under $d_R$ provide the first explicit sub-polynomial bound for high-dimensional, non-linear functionals of Gaussian processes beyond the i.i.d. setting. Building on this, we derive explicit Berry-Esseen bounds under both $d_R$ and $d_{\mathscr{C}}$ for multiple statistical examples, such as the method of moments, empirical characteristic functions, empirical moment-generating functions, and functional limit theorems in high-dimensional settings. |
| title | Quantitative bounds for high-dimensional non-linear functionals of Gaussian processes |
| topic | Probability 60F05, 60G15, 60H07 |
| url | https://arxiv.org/abs/2502.17718 |