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<p>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&#xa0;0 or&#xa0;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.</p>

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Zero–One Laws for Events with Positional Symmetries

  • 期刊论文

  • 发表日期：

    2025年3月24日

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  • Yahya Ayach,
  • Anthony Khairallah,
  • Tia Manoukian,
  • Jad Mchaimech,
  • Adam Salha,
  • Siamak Taati

摘要

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.