Functional central limit theorem for topological functionals of Gaussian critical points
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908709729337344 |
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| author | Hirsch, Christian Lachièze-Rey, Raphaël |
| author_facet | Hirsch, Christian Lachièze-Rey, Raphaël |
| contents | We consider Betti numbers of the excursion of a smooth Euclidean Gaussian field restricted to a rectangular window, in the asymptotics where the window grows to R^d . With motivations coming from Topological Data Analysis, we derive a functional Central Limit Theorem where the varying argument is the thresholding parameter, under assumptions of regularity and covariance decay for the field and its derivatives. We also show fixed-level CLTs coming from martingale based techniques inspired from the theory of geometric stabilisation, and limiting non-degenerate variance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11429 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Functional central limit theorem for topological functionals of Gaussian critical points Hirsch, Christian Lachièze-Rey, Raphaël Probability We consider Betti numbers of the excursion of a smooth Euclidean Gaussian field restricted to a rectangular window, in the asymptotics where the window grows to R^d . With motivations coming from Topological Data Analysis, we derive a functional Central Limit Theorem where the varying argument is the thresholding parameter, under assumptions of regularity and covariance decay for the field and its derivatives. We also show fixed-level CLTs coming from martingale based techniques inspired from the theory of geometric stabilisation, and limiting non-degenerate variance. |
| title | Functional central limit theorem for topological functionals of Gaussian critical points |
| topic | Probability |
| url | https://arxiv.org/abs/2411.11429 |