nLab zero-one law

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Contents

Idea

In probability theory, zero-one laws are conditions for which a certain probability measure can only assume the values zero and one (i.e. it is a zero-one measure).

Sometimes they can be seen as conditions for which a probability measure, usually an iid one, is ergodic (that is, it is zero-one on the invariant sets of some action).

Statements

Hewitt-Savage zero-one law

Let (X,p)(X,p) be a probability space. Consider the infinite product X X^\mathbb{N}, and form the iid measure p˜\tilde{p} on X X^\mathbb{N} whose marginals are given by pp.

Then for every exchangeable event AX A\subseteq X^\mathbb{N} we have that either p˜(A)=0\tilde{p}(A)=0 or p˜(A)=1\tilde{p}(A)=1. That is, p˜\tilde{p} restricted to the exchangeable sigma-algebra is a zero-one measure.

(The equivalent characterizations of zero-one measures give equivalent formulations of this statement.)

Kolmogorov’s zero-one law

Let (X,p)(X,p) be a probability space. Consider the infinite product X X^\mathbb{N}, and form the iid measure p˜\tilde{p} on X X^\mathbb{N} whose marginals are given by pp.

Then for every tail event AX A\subseteq X^\mathbb{N} we have that either p˜(A)=0\tilde{p}(A)=0 or p˜(A)=1\tilde{p}(A)=1. That is, p˜\tilde{p} restricted to the tail sigma-algebra is a zero-one measure.

(The equivalent characterizations of zero-one measures give equivalent formulations of this statement.)

Levy’s zero-one law

See martingale convergence theorem?.

In categorical probability

Markov categories

(For now, see Fritz-Rischel’20.)

Dagger categories

(For now, see Ensarguet-Perrone’23.)

See also

References

  • Tobias Fritz and Eigil Fjeldgren Rischel, Infinite products and zero-one laws in categorical probability, Compositionality 2(3) 2020. (arXiv:1912.02769)

  • Noé Ensarguet, Paolo Perrone, Categorical probability spaces, ergodic decompositions, and transitions to equilibrium, arXiv:2310.04267

category: probability

Last revised on July 19, 2024 at 18:33:39. See the history of this page for a list of all contributions to it.