A composite generalization of Ville’s martingale theorem using e-processes

Ruf, Johannes ORCID: 0000-0003-3616-2194, Larsson, Martin, Koolen, Wouter m. and Ramdas, Aaditya (2023) A composite generalization of Ville’s martingale theorem using e-processes. Electronic Journal of Probability, 28. ISSN 1083-6489

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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.

Item Type:	Article
Official URL:	https://projecteuclid.org/journals/electronic-jour...
Additional Information:	© 2023, Institute of Mathematical Statistics.
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics
Date Deposited:	22 Nov 2023 08:57
Last Modified:	18 Nov 2024 23:09
URI:	http://eprints.lse.ac.uk/id/eprint/120825

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