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

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.

💡 Research Summary

The paper establishes a new Fuk‑Nagaev inequality for the maximal norm of martingale sequences taking values in a (2,D)-smooth Banach space, under the assumption that a finite number of conditional moments exist. The authors combine two powerful techniques: Rio’s (2017) optimization of a Chernoff bound for truncated martingales, and Pinelis’s (1994) exponential moment bound for norms in smooth Banach spaces. By merging these tools they obtain a concentration inequality that mirrors the classical real‑valued Fuk‑Nagaev result, but without any centering term and with explicit constants that depend only on the smoothness parameter D and the moment order q > 2.

Main Result (Theorem 2.2).
Let (X) be a (2,D)-smooth Banach space and ((M_i){i=0}^n) a martingale with differences (\xi_i = M_i-M{i-1}). Assume
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