The Filtration of the split-words process

TL;DR

This paper characterizes when filtrations of split-word processes are standard or non-standard in discrete negative time, providing a practical criterion and illustrating how time acceleration can alter standardness.

Contribution

It proves Laurent's sufficient condition for non-standardness is also necessary, establishing a definitive standardness criterion for these filtrations.

Findings

Laurent's condition is both necessary and sufficient for non-standardness

A non-standard filtration can become standard when time is accelerated

Provides a practical criterion for determining filtration standardness

Abstract

M. Smorodinsky and S. Laurent have initiated the study of the filtrations of split-word processes, in the framework of discrete negative time. For these filtrations, we show that Laurent's sufficient condition for non standardness is also necessary, thus yielding a practical standardness criterion. In turn, this criterion enables us to exhibit a non standard filtration which becomes standard when time is accelerated by omitting infinitely many instants of time.