Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60G10 Stationary processes
- 28Dxx Measure-theoretic ergodic theory, see also {11K50, 11K55, 22D40, 47A35, 54H20, 58Fxx, 60Fxx, 60G10}

**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*