Suficient conditions of standardness for filtrations of stationary processes taking values in a finite space

TL;DR

This paper extends previous results on the standardness of filtrations for stationary processes with finite state spaces, providing new sufficient conditions based on average gaps rather than maximal gaps.

Contribution

It generalizes prior work from binary to finite state spaces and introduces conditions based on average gaps for standardness of filtrations.

Findings

Standardness conditions are extended to finite state spaces.

Average gap-based criteria replace maximal gap conditions.

Results apply to a broader class of stationary processes.

Abstract

Let $X$ be a stationary process with finite state-space $A$. Bressaud et al. recently provided a sufficient condition for the natural filtration of $X$ to be standard when $A$ has size 2. Their condition involves the conditional laws $p(\cdot|x)$ of $X_0$ conditionally on the whole past $(X_k)_{k \le -1}=x$ and controls the strength of the influence of the "old" past of the process on its present $X_0$. It involves the maximal gaps between $p(\cdot|x)$ and $p(\cdot|y)$ for infinite sequences $x$ and $y$ which coincide on their $n$ last terms. In this paper, we first show that a slightly stronger result holds for any finite state-space. Then, we provide sufficient conditions for standardness based on average gaps instead of maximal gaps.