Sufficient conditions for the filtration of a stationary processes to be standard

TL;DR

This paper extends previous results by providing new sufficient conditions under which the natural filtration of a stationary process is standard, generalizing from finite to more complex state spaces using advanced coupling techniques.

Contribution

It introduces a novel coupling method for laws with densities relative to a measure, broadening the criteria for the standardness of filtrations in stationary processes.

Findings

Established new sufficient conditions for standardness in general state spaces

Developed a complex coupling construction for laws with densities

Generalized previous finite-state results to broader settings

Abstract

Let $X$ be a stationary process with values in some $\sigma$-finite measured state space $(E,\mathcal{E},\pi)$, indexed by ${\mathbb Z}$. Call ${\mathcal F}^X$ its natural filtration. In \cite{ceillierstationary}, sufficient conditions were given for ${\mathcal F}^X$ to be standard when $E$ is finite. The proof of this result used a coupling of all probabilities on the finite set $E$. In this paper, we construct a coupling of all laws having a density with regard to $\pi$, which is much more involved. Then, we provide sufficient conditions for ${\mathcal F}^X$ to be standard, generalizing those in \cite{ceillierstationary}.