Levy's zero-one law in game-theoretic probability

TL;DR

This paper establishes a game-theoretic version of Levy's zero-one law, deriving several classical probabilistic results and exploring foundational concepts in game-theoretic probability theory.

Contribution

It introduces a game-theoretic formulation of Levy's zero-one law and derives related classical probabilistic results within this framework.

Findings

Proves a game-theoretic version of Levy's zero-one law

Derives non-stochastic Kolmogorov's zero-one law and ergodicity of Bernoulli shifts

Establishes a zero-one law for dependent trials

Abstract

We prove a game-theoretic version of Levy's zero-one law, and deduce several corollaries from it, including non-stochastic versions of Kolmogorov's zero-one law, the ergodicity of Bernoulli shifts, and a zero-one law for dependent trials. Our secondary goal is to explore the basic definitions of game-theoretic probability theory, with Levy's zero-one law serving a useful role.