Fuk-Nagaev inequality in smooth Banach spaces: Optimum bounds for distributions of heavy-tailed martingales
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918247543078912 |
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| author | Mollenhauer, Mattes Fiedler, Christian |
| author_facet | Mollenhauer, Mattes Fiedler, Christian |
| contents | We derive a Fuk-Nagaev inequality for the maxima of norms of martingale sequences in smooth Banach spaces which allow for a finite number of higher conditional moments. The bound is obtained by combining an optimization approach for a Chernoff bound due to Rio (2017) with a classical bound for moment generating functions of smooth Banach space norms by Pinelis (1994). Our result improves comparable infinite-dimensional bounds in the literature by removing unnecessary centering terms and giving precise constants. As an application, we propose a McDiarmid-type bound for vector-valued functions which allow for a uniform bound on their conditional higher moments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_10012 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fuk-Nagaev inequality in smooth Banach spaces: Optimum bounds for distributions of heavy-tailed martingales Mollenhauer, Mattes Fiedler, Christian Probability 60G42, 60E15 We derive a Fuk-Nagaev inequality for the maxima of norms of martingale sequences in smooth Banach spaces which allow for a finite number of higher conditional moments. The bound is obtained by combining an optimization approach for a Chernoff bound due to Rio (2017) with a classical bound for moment generating functions of smooth Banach space norms by Pinelis (1994). Our result improves comparable infinite-dimensional bounds in the literature by removing unnecessary centering terms and giving precise constants. As an application, we propose a McDiarmid-type bound for vector-valued functions which allow for a uniform bound on their conditional higher moments. |
| title | Fuk-Nagaev inequality in smooth Banach spaces: Optimum bounds for distributions of heavy-tailed martingales |
| topic | Probability 60G42, 60E15 |
| url | https://arxiv.org/abs/2512.10012 |