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 [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].
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Gaël Ceillier, Christophe Leuridan. Sufficient conditions for the filtration of a stationary processes to be standard. Probability Theory and Related Fields, Springer Verlag, 2017, 167 (3-4), pp.979-999. ⟨10.1007/s00440-016-0696-2⟩. ⟨hal-01264055⟩

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