Fuk-Nagaev inequality in smooth Banach spaces: Optimum bounds for distributions of heavy-tailed martingales

Mattes Mollenhauer; Christian Fiedler

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arXiv · 2512.10012

Fuk-Nagaev inequality in smooth Banach spaces: Optimum bounds for distributions of heavy-tailed martingales

Mattes Mollenhauer·Christian Fiedler

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Abstract

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.

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BibTeX RIS

Mattes Mollenhauer, Christian Fiedler. 2025-12-10. Fuk-Nagaev inequality in smooth Banach spaces: Optimum bounds for distributions of heavy-tailed martingales. https://arxiv.org/abs/2512.10012

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