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

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 Ceillier (Ann Probab 40(5):1980–2007, 2012), sufficient conditions were given for ${{\mathcal {F}}}^X$ to be standard when E is finite, and the proof 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 Ceillier (Ann Probab 40(5):1980–2007, 2012).