Filtrations at the threshold of standardness

• 成果类型: Article
• 署名作者: Ceillier, Gael; Leuridan, Christophe
• 署名单位: Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-013-0496-x
• 发表日期: 2014
• 页码: 785-808

摘要

A. Vershik discovered that filtrations indexed by the non-positive integers may have a paradoxical asymptotic behaviour near the time , called non-standardness. For example, two dyadic filtrations with trivial tail -field are not necessarily isomorphic. Yet, from any essentially separable filtration indexed by the non-positive integers, one can extract a subsequence which is a standard filtration. In this paper, we focus on the non-standard filtrations which become standard if (and only if) infinitely many integers are skipped. We call them filtrations at the threshold of standardness, since they are as close to standardardness as they can be although they are non-standard. Two classes of filtrations are studied, first the filtrations of the split-words processes, second some filtrations inspired by an unpublished example of B. Tsirelson. They provide examples which disprove some naive intuitions. For example, it is possible to have a standard filtration extracted from a non-standard one with no intermediate (for extraction) filtration at the threshold of standardness. It is also possible to have a filtration which provides a standard filtration on the even times but a non-standard filtration on the odd times.