nLab zero-one law

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

Then for every exchangeable event $A\subseteq X^\mathbb{N}$ we have that either $\tilde{p}(A)=0$ or $\tilde{p}(A)=1$. That is, $\tilde{p}$ 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 $(X,p)$ be a probability space. Consider the infinite product $X^\mathbb{N}$, and form the iid measure $\tilde{p}$ on $X^\mathbb{N}$ whose marginals are given by $p$.

Then for every tail event $A\subseteq X^\mathbb{N}$ we have that either $\tilde{p}(A)=0$ or $\tilde{p}(A)=1$. That is, $\tilde{p}$ 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

• probability theory, categorical probability

• zero-one measure, zero-one kernel, deterministic random variable, Dirac measure

• ergodicity, invariant measure, invariant set, ergodic decomposition theorem

• de Finetti's theorem, exchangeability, tail event

• law of large numbers, ergodic theorem?

• Markov category, probability monad

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