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Theses

Filtrations à temps discret

Abstract : Standardness is an important invariant in the theory of filtrations indexed by negative integer times. The main purpose of this thesis is to determine whether some filtrations are standard or not. We first focus on the filtrations of split-word processes, introduced and studied by Smorodinsky and by Laurent. We prove that Laurent's sufficient condition for non standardness is also necessary. This yields a practical criterion of standardness. In turn, this criterion enables us to exhibit non standard filtrations which become standard when time is accelerated by omitting infinitely many instants of time. Secondly, we study the natural filtrations of stationary processes on finite state-spaces. Recently Bressaud et al.\ provided a sufficient condition for the natural filtration of such a process $(X_k)_k$ to be standard when the state-space has size $2$. Their condition involves the conditional laws $p(\cdot|x)$ of $X_0$ conditionally on $(X_k)_{k \le -1}=x$ and controls the influence of the remote past of the process on its present $X_0$. Bressaud and al.\ measure the maximal strength of this influence. We provide sufficient conditions for standardness based on some average gaps between these conditional laws, instead of the maximal gaps.
Keywords : Filtrations
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Contributor : Gael Ceillier <>
Submitted on : Tuesday, February 1, 2011 - 12:28:26 PM
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Gael Ceillier. Filtrations à temps discret. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2010. Français. ⟨tel-00561467⟩

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