Filtrations of random processes in the light of classification theory. I. A topological zeroone law
Abstract
Filtered probability spaces (called "filtrations" for short) are shown to satisfy such a topological zeroone 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 Gspaces. The set of all isomorphic classes of filtrations may be identified with the orbit space X/G for a special Polish Gspace X. A "property of filtrations" means a Ginvariant subset of X having the Baire property. "Almost all filtrations" means a comeager subset of X (the Baire category approach). The zeroone 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 July 2001
 arXiv:
 arXiv:math/0107121
 Bibcode:
 2001math......7121T
 Keywords:

 Mathematics  Probability;
 Mathematics  Logic;
 60G07;
 03E15
 EPrint:
 35 pages