A Caveat on Metrizing Convergence in Distribution on Hilbert Spaces
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915765490286592 |
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| author | Bassetti, Federico Bourguin, Solesne Campese, Simon Peccati, Giovanni |
| author_facet | Bassetti, Federico Bourguin, Solesne Campese, Simon Peccati, Giovanni |
| contents | We consider Sobolev-type distances on probability measures over separable Hilbert spaces involving the Schatten-$p$ norms, which include as special cases a distance first introduced by Bourguin and Campese (2020) when $p=2$, and a distance introduced by Giné and Leon (1980) when $p=\infty$. Our analysis shows that, unless $p=\infty$, these distances fail to metrize convergence in distribution in infinite dimensions. This clarifies several inconsistencies and misconceptions in the recent literature that arose from confusion between different types of distances. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13427 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Caveat on Metrizing Convergence in Distribution on Hilbert Spaces Bassetti, Federico Bourguin, Solesne Campese, Simon Peccati, Giovanni Probability We consider Sobolev-type distances on probability measures over separable Hilbert spaces involving the Schatten-$p$ norms, which include as special cases a distance first introduced by Bourguin and Campese (2020) when $p=2$, and a distance introduced by Giné and Leon (1980) when $p=\infty$. Our analysis shows that, unless $p=\infty$, these distances fail to metrize convergence in distribution in infinite dimensions. This clarifies several inconsistencies and misconceptions in the recent literature that arose from confusion between different types of distances. |
| title | A Caveat on Metrizing Convergence in Distribution on Hilbert Spaces |
| topic | Probability |
| url | https://arxiv.org/abs/2509.13427 |