Zero–One Laws for Events with Positional Symmetries

Yahya Ayach, Anthony Khairallah, Tia Manoukian, Jad Mchaimech, Adam Salha and Siamak Taati ()
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Yahya Ayach: American University of Beirut
Anthony Khairallah: American University of Beirut
Tia Manoukian: American University of Beirut
Jad Mchaimech: American University of Beirut
Adam Salha: American University of Beirut
Siamak Taati: American University of Beirut

Journal of Theoretical Probability, 2025, vol. 38, issue 2, 1-20

Abstract: 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: Zero-one laws; Symmetries; Information inequalities; Exchangeability; Ergodicity; Random graphs; Simple renormalization; 60F20; 60G09; 94A15 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-025-01411-2

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