Zero-one laws for events with positional symmetries

TL;DR

This paper proves that events with certain symmetries in infinite i.i.d. or similar processes are almost surely deterministic, extending classical zero-one laws to broader contexts using an information-theoretic approach.

Contribution

It introduces a unified information-theoretic proof that generalizes zero-one laws for symmetric events beyond traditional i.i.d. settings.

Findings

Generalizes Hewitt-Savage zero-one law

Applies to infinite random graphs and renormalization processes

Uses an information-theoretic argument for the proof

Abstract

We use an information-theoretic argument due to O'Connell (2000) to prove that every sufficiently symmetric event concerning a countably infinite family of independent and identically distributed random variables is deterministic (i.e., has a probability of either 0 or 1). The i.i.d. condition can be relaxed. This result encompasses the Hewitt-Savage zero-one law and the ergodicity of the Bernoulli process, but also applies to other scenarios such as infinite random graphs and simple renormalization processes.