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.

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Electronic Journal of Probability

Ruf, J., Larsson, M., Koolen-Wijkstra, W., & Ramdas, A. (2023). A composite generalization of Ville’s martingale theorem using e-processes. Electronic Journal of Probability, 28, 1–21. doi:10.1214/23-EJP1019