Convergence of cones of metric measure spaces and its application to Cauchy distribution
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914689144848384 |
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| author | Esaki, Syota Kazukawa, Daisuke Mitsuishi, Ayato |
| author_facet | Esaki, Syota Kazukawa, Daisuke Mitsuishi, Ayato |
| contents | We prove that the sequence of cones of metric measure spaces converges if the sequence of base spaces converges in Gromov's box, concentration, and weak topologies. As an application, we show that the generalized Cauchy distribution with suitable scaling converges to a half line in the concentration topology as the dimension diverges to infinity. This is a new example distinguished from previously known examples such as Gaussian distributions and typical closed Riemannian manifolds with constant Ricci curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14331 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence of cones of metric measure spaces and its application to Cauchy distribution Esaki, Syota Kazukawa, Daisuke Mitsuishi, Ayato Metric Geometry Probability 53C23, 60B12, 53C21, 60A10 We prove that the sequence of cones of metric measure spaces converges if the sequence of base spaces converges in Gromov's box, concentration, and weak topologies. As an application, we show that the generalized Cauchy distribution with suitable scaling converges to a half line in the concentration topology as the dimension diverges to infinity. This is a new example distinguished from previously known examples such as Gaussian distributions and typical closed Riemannian manifolds with constant Ricci curvature. |
| title | Convergence of cones of metric measure spaces and its application to Cauchy distribution |
| topic | Metric Geometry Probability 53C23, 60B12, 53C21, 60A10 |
| url | https://arxiv.org/abs/2402.14331 |