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

International audienceLet $X$ 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 [3], 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 [3]

On standardness and I-cosiness

peer reviewedThe object of study of this work is the invariant characteristics of filtrations in discrete, negative time, pioneered by Vershik. We prove the equivalence between I-cosiness and standardness without using Vershik’s standardness criterion. The equivalence between I-cosiness and productness for homogeneous filtrations is further investigated by showing that the I-cosiness criterion is equivalent to Vershik’s first level criterion separately for each random variable. We also aim to derive the elementary properties of both these criteria, and to give a survey and some complements on the published and unpublished literature