Large Sample Theory for Bures-Wasserstein Barycentres
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866929574883885056 |
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| author | Santoro, Leonardo V. Panaretos, Victor M. |
| author_facet | Santoro, Leonardo V. Panaretos, Victor M. |
| contents | We establish a strong law of large numbers and a central limit theorem in the Bures-Wasserstein space of covariance operators -- or equivalently centred Gaussian measures -- over a general separable Hilbert space. Specifically, we show that empirical barycentre sequences indexed by sample size are almost certainly relatively compact, with accumulation points comprising population barycentres. We give a sufficient regularity condition for the limit to be unique. When the limit is unique, we also establish a central limit theorem under a refined pair of moment and regularity conditions.
Finally, we prove strong operator convergence of the empirical optimal transport maps to their population counterparts. Though our results naturally extend finite-dimensional counterparts, including associated regularity conditions, our techniques are distinctly different owing to the functional nature of the problem in the general setting. A key element is the characterisation of compact sets in the Bures-Wasserstein topology that reflects an ordered Heine-Borel property of the Bures-Wasserstein space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_15592 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Large Sample Theory for Bures-Wasserstein Barycentres Santoro, Leonardo V. Panaretos, Victor M. Probability Statistics Theory 60B12, 60G57, 60H25, 62R20, 62R30 We establish a strong law of large numbers and a central limit theorem in the Bures-Wasserstein space of covariance operators -- or equivalently centred Gaussian measures -- over a general separable Hilbert space. Specifically, we show that empirical barycentre sequences indexed by sample size are almost certainly relatively compact, with accumulation points comprising population barycentres. We give a sufficient regularity condition for the limit to be unique. When the limit is unique, we also establish a central limit theorem under a refined pair of moment and regularity conditions. Finally, we prove strong operator convergence of the empirical optimal transport maps to their population counterparts. Though our results naturally extend finite-dimensional counterparts, including associated regularity conditions, our techniques are distinctly different owing to the functional nature of the problem in the general setting. A key element is the characterisation of compact sets in the Bures-Wasserstein topology that reflects an ordered Heine-Borel property of the Bures-Wasserstein space. |
| title | Large Sample Theory for Bures-Wasserstein Barycentres |
| topic | Probability Statistics Theory 60B12, 60G57, 60H25, 62R20, 62R30 |
| url | https://arxiv.org/abs/2305.15592 |