Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

Markov chains with positive transitions are not determined by any p-marginals

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Résumé: We prove that if $X=(X_n)_{n\in Z}$ is a finite valued stationary Markov Chain with strictly positive probability transitions, then for any natural number $p$, there exists a continuum of finite valued non Markovian processes which have the $p$-marginal distributions of $X$ and with positive entropy, whereas for an irrational rotation $R=R_{\alpha}$ and essentially bounded real measurabe function $f$ with no zero Fourier coefficient on the unit circle with normalized Lebesgue measure, the process $(f(R^n))_{n\in Z}$ is uniquely determined by its three-dimensional distributions in the class of ergodic processes. We give also a family of Gaussian non-Markovian dynamical systems for which the symbolic dynamic associated to the time zero partition has the two-dimensional distributions of a reversible mixing Markov Chain.

Mots Clés: Finite dimensional distributions ; ergodic Markov Chain ; mixing Gaussian dynamical system ; entropy ; group rotation

Date: 2000-01-27

Prépublication numéro: PMA-561