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arXiv · 2406.14902

Zero-one laws for events with positional symmetries

Yahya Ayach·Anthony Khairallah·Tia Manoukian·Jad Mchaimech·Adam Salha·Siamak Taati
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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.

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Yahya Ayach, Anthony Khairallah, Tia Manoukian, Jad Mchaimech, Adam Salha, Siamak Taati. 2024-06-21. Zero-one laws for events with positional symmetries. https://doi.org/10.1007/s10959-025-01411-2

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