Central Limit Theorem for Random Partial Sphere Coverings in High Dimensions
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910115230121984 |
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| author | Hoehner, Steven Thäle, Christoph |
| author_facet | Hoehner, Steven Thäle, Christoph |
| contents | We study a random partial covering model on the $(d-1)$-dimensional unit sphere, where $N$ spherical caps are placed independently and uniformly at random, each covering a surface fraction of $1/N$. This model provides a continuous geometric analogue of the classical balls-into-bins problem. We establish a Central Limit Theorem for the volume of the resulting random partial covering, showing that its fluctuations are asymptotically Gaussian. Moreover, we obtain a quantitative bound on the rate of convergence in the Kolmogorov distance. Our results hold both in fixed dimension and in a high-dimensional regime where the dimension grows at most logarithmically with $N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_07711 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Central Limit Theorem for Random Partial Sphere Coverings in High Dimensions Hoehner, Steven Thäle, Christoph Probability Metric Geometry 60D05 (Primary) 60F05 (Secondary) We study a random partial covering model on the $(d-1)$-dimensional unit sphere, where $N$ spherical caps are placed independently and uniformly at random, each covering a surface fraction of $1/N$. This model provides a continuous geometric analogue of the classical balls-into-bins problem. We establish a Central Limit Theorem for the volume of the resulting random partial covering, showing that its fluctuations are asymptotically Gaussian. Moreover, we obtain a quantitative bound on the rate of convergence in the Kolmogorov distance. Our results hold both in fixed dimension and in a high-dimensional regime where the dimension grows at most logarithmically with $N$. |
| title | Central Limit Theorem for Random Partial Sphere Coverings in High Dimensions |
| topic | Probability Metric Geometry 60D05 (Primary) 60F05 (Secondary) |
| url | https://arxiv.org/abs/2604.07711 |