A composite generalization of Ville's martingale theorem

TL;DR

This paper extends Ville's martingale theorem to a composite setting using e-processes and introduces a new line-crossing inequality to prove a generalized strong law of large numbers, linking measure-zero events to martingale behavior.

Contribution

It provides a novel composite version of Ville's theorem, introduces a new inequality for sums of finite first moment variables, and proves a generalized SLLN with measure-zero violation events.

Findings

E-processes explode to infinity on measure-zero violating sequences.

A new line-crossing inequality for sums with finite first moments.

Generalized SLLN with violations of measure zero.

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.