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

arXiv: https://arxiv.org/abs/2203.04485 · HTML: https://mlsanity.com/paper/2203.04485/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Mar 2022.

## Built on by (most cited first, 10 of 10)

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- [Positive Semidefinite Matrix Supermartingales](https://mlsanity.com/paper/2401.15567/) (2024)
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## Abstract (arXiv)

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.