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).
Let be a probability space. Consider the infinite product , and form the iid measure on whose marginals are given by .
Then for every exchangeable event we have that either or . That is, restricted to the exchangeable sigma-algebra is a zero-one measure.
(The equivalent characterizations of zero-one measures give equivalent formulations of this statement.)
Let be a probability space. Consider the infinite product , and form the iid measure on whose marginals are given by .
Then for every tail event we have that either or . That is, restricted to the tail sigma-algebra is a zero-one measure.
(The equivalent characterizations of zero-one measures give equivalent formulations of this statement.)
See martingale convergence theorem?.
(For now, see Fritz-Rischel’20.)
(For now, see Ensarguet-Perrone’23.)
zero-one measure, zero-one kernel, deterministic random variable, Dirac measure
ergodicity, invariant measure, invariant set, ergodic decomposition theorem
law of large numbers, ergodic theorem?
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
Last revised on July 19, 2024 at 18:33:39. See the history of this page for a list of all contributions to it.