Filtrations of random processes in the light of classification theory. I. A topological zero-one law

Details Filtrations of random processes in the light of classification theory. I. A topological zero-one law Filtrations of random processes in the light of classification theory. I. A topological zero-one law

2001-07-17

arxiv.org/abs/math/0107121v1

Mathematics

Probability

35 pages

Scientific paper

Filtered probability spaces (called "filtrations" for short) are shown to satisfy such a topological zero-one law: for every property of filtrations, either the property holds for almost all filtrations, or its negation does. In particular, almost all filtrations are conditionally nonatomic. An accurate formulation is given in terms of orbit equivalence relations on Polish G-spaces. The set of all isomorphic classes of filtrations may be identified with the orbit space X/G for a special Polish G-space X. A "property of filtrations" means a G-invariant subset of X having the Baire property. "Almost all filtrations" means a comeager subset of X (the Baire category approach). The zero-one law is a kind of ergodicity of X. It holds for filtrations both in discrete and continuous time. The interplay between probability theory and descriptive set theory could be interesting for both parties.

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