Filtrations at the threshold of standardness

TL;DR

This paper investigates filtrations that are nearly standard, called at the threshold of standardness, exploring their properties and providing examples that challenge naive assumptions about their structure.

Contribution

It introduces the concept of filtrations at the threshold of standardness and studies specific classes, including split-words and Tsirelson-inspired filtrations, revealing surprising behaviors.

Findings

Existence of filtrations at the threshold of standardness that are non-standard but become standard when infinitely many indices are skipped.

Examples of filtrations with mixed standard and non-standard behavior on even and odd times.

Disproof of naive intuitions about the structure and extraction of standard filtrations from non-standard ones.

Abstract

A. Vershik discovered that filtrations indexed by the non-positive integers may have a paradoxical asymptotic behaviour near the time $-\infty$, called non-standardness. For example, two dyadic filtrations with trivial tail $\sigma$-field are not necessarily isomorphic. Yet, any essentially separable filtration indexed by the non-positive integers becomes standard when sufficiently many integers are skipped. 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 class of filtrations are studied, first the filtrations of the split-words processes, second some filtrations inspired by an unpublished example of B. Tsirelson.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 disproves 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.