Zero-one laws for events with positional symmetries
Probability
2025-03-26 v2 Information Theory
math.IT
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.
Keywords
Cite
@article{arxiv.2406.14902,
title = {Zero-one laws for events with positional symmetries},
author = {Yahya Ayach and Anthony Khairallah and Tia Manoukian and Jad Mchaimech and Adam Salha and Siamak Taati},
journal= {arXiv preprint arXiv:2406.14902},
year = {2025}
}
Comments
16 pages, 4 figures