The Coincidence of Measure Algebras Under an Exchangeable Probability
Abstract
This note is concerned with countably infinite product sigma-fields and their invariant, tail, and exchangeable sub-sigma-fields. Under an exchangeable probability the three sub-sigma-fields coincide as measure algebras (the theorems (1) and (7)). An immediate consequence is the Hewitt-Savage 0-1 law. A later section includes examples which by and large preclude extensions of (1) and (7) to probabilities merely invariant under the shift. However, at least one interesting conjecture of David Freedman remains to be settled. The results presented here serve to clarify and extend a remark by Halmos about power product probabilities. They also extend a theorem set forth by Meyer to the effect that in a unilateral countable product space, under an exchangeable probability, exchangeable and tail cr-fields coincide as measure algebras. The final section contains the answer to a question posed in the paper by Chung and Doob.
Document Details
- Document Type
- Technical Report
- Publication Date
- Feb 10, 1970
- Accession Number
- ADA594296
Entities
People
- Richard A. Olshen
Organizations
- Stanford University