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September 2012 Sufficient conditions of standardness for filtrations of stationary processes taking values in a finite space
Gaël Ceillier
Ann. Probab. 40(5): 1980-2007 (September 2012). DOI: 10.1214/11-AOP666

Abstract

Let $X$ be a stationary process with finite state-space $A$. Bressaud et al. [Ann. Probab. 34 (2006) 1589–1600] 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 $p(\cdot|x)$ of $X_{0}$ conditionally on the whole past $(X_{k})_{k\le-1}=x$ and controls the strength of the influence of the “old” past of the process on its present $X_{0}$. It involves the maximal gaps between $p(\cdot|x)$ and $p(\cdot|y)$ 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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Gaël Ceillier. "Sufficient conditions of standardness for filtrations of stationary processes taking values in a finite space." Ann. Probab. 40 (5) 1980 - 2007, September 2012. https://doi.org/10.1214/11-AOP666

Information

Published: September 2012
First available in Project Euclid: 8 October 2012

zbMATH: 1266.60064
MathSciNet: MR3025707
Digital Object Identifier: 10.1214/11-AOP666

Subjects:
Primary: 60A99

Keywords: filtration , non-Markovian processes , parametrization

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 5 • September 2012
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