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A composite generalization of Ville's martingale theorem ...
[Submitted on 9 Mar 2022 (v1), last revised 28 Aug 2026 (this ve · 2022-03-09 · via math.ST updates on arXiv.org

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Abstract:We provide a composite version of Ville's theorem that an event has zero measure if and only if there exists a nonnegative martingale which explodes to infinity when that event occurs. This is a classic result connecting measure-theoretic probability to the sequence-by-sequence game-theoretic probability, recently developed by Shafer and Vovk. Our extension of Ville's result involves appropriate composite generalizations of nonnegative martingales and measure-zero events: these are respectively provided by ``e-processes'', and a new inverse capital outer measure. We then develop a novel line-crossing inequality for sums of random variables which are only required to have a finite first moment, which we use to prove a composite version of the strong law of large numbers (SLLN). This allows us to show that violation of the SLLN is an event of outer measure zero and that our e-process explodes to infinity on every such violating sequence, while this is provably not achievable with a nonnegative (super)martingale.

Submission history

From: Johannes Ruf [view email]
[v1] Wed, 9 Mar 2022 02:02:35 UTC (27 KB)
[v2] Wed, 3 May 2023 19:06:07 UTC (33 KB)
[v3] Fri, 28 Aug 2026 21:52:40 UTC (33 KB)