Library Header Image
LSE Research Online LSE Library Services

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

[img] Text (Ruf_composite-generalization--published) - Published Version
Available under License Creative Commons Attribution.

Download (463kB)

Identification Number: 10.1214/23-EJP1019


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:
Additional Information: © 2023, Institute of Mathematical Statistics.
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 22 Nov 2023 08:57
Last Modified: 15 Jul 2024 02:42

Actions (login required)

View Item View Item


Downloads per month over past year

View more statistics