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

2011/01/10 by Gaël Ceillier, Ceillier, Gaël
Computer Science · Engineering · Mathematics · #Advanced Data Processing Techniques #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Control Systems and Analysis #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1101.1931

openalex publication_date 2011/01/10 · arxiv created 2011/10/25 · arxiv updated 2015/03/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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(⋅|x) of X₀ conditionally on the whole past (X_k)_{k ≤ -1}=x and controls the strength of the influence of the "old" past of the process on its present X₀. It involves the maximal gaps between p(⋅|x) and p(⋅|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.

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