Zero-one laws for events with positional symmetries

TL;DR

The paper develops a unifying zero-one law for events governed by a countable family of random variables under positional symmetries, proving that sufficient injective symmetries force such events to have probability in {0,1}. Building on O'Connell's information-theoretic approach, the authors first establish a general iid result and then extend it to non-iid settings via approximate independence and closeness of distributions, yielding robust corollaries. The framework encompasses classical Hewitt–Savage and shift-ergodicity results and applies to infinite random graphs and renormalization maps, with further reach to Kolmogorov-type mixing settings and other dependent structures. Overall, the work provides a versatile, information-theoretic toolkit for deterministic behavior arising from symmetry in broad stochastic systems.

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.

Figures (4)

• Figure 1: An intuitive illustration of O'Connell's inequality. Each region represents the information carried by one of the random variables. The regions corresponding to independent random variables are depicted to be disjoint. This illustration however must not be taken too literally because, although $W_1,W_2,\ldots,W_n$ are independent, the information pieces they carry about $V$ need not be independent.
• Figure 2: A few iterations of the recursive construction from Example \ref{['exp:graph:with-symmetries']} with arbitrary choices. The dashed rectangles indicate the graphs $G_0$, $G_2$, …, $G_5$ included in one another.
• Figure 3: The renormalization process of Example \ref{['exp:renormalization:majority']}.
• Figure 4: The dynamics of $p\mapsto f(p)$ in Example \ref{['exp:renormalization:majority']}. As $n\to\infty$, we have $f^n(p)\to 1$ if $p>1/2$ and $f^n(p)\to 0$ if $p<1/2$.

Theorems & Definitions (30)

• Theorem A: Hewitt-Savage zero-one law
• Theorem B: Ergodicity of the Bernoulli process
• Definition 1: Positional symmetry
• Theorem 1: Zero-one law for iid RVs and symmetric events
• Theorem 2: Zero-one law for independent but not identically distributed RVs
• Theorem 3: Zero-one law for asymptotically independent RVs
• Lemma A: O'Connell's inequality
• Lemma B: Closedness of independence
• Remark 1: Information-theoretic "proof' of the Kolmogorov zero-one law
• proof : Proof of Theorem \ref{['thm:main:iid']}