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Shannon entropy for stationary processes and dynamical systems

Published online by Cambridge University Press:  01 April 2008

D. HAMDAN ,
W. PARRY  and
J.-P. THOUVENOT
D. HAMDAN
Affiliation:
Laboratoire de Probabilités et Modèles aléatoires, UMR 7599, Universités Paris 6 et Paris 7, Boǐte Courrier 188, 4 Place Jussieu, 75252 Paris cedex 05, France (email: hamdan@ccr.jussieu.fr)
W. PARRY
Affiliation:
Laboratoire de Probabilités et Modèles aléatoires, UMR 7599, Universités Paris 6 et Paris 7, Boǐte Courrier 188, 4 Place Jussieu, 75252 Paris cedex 05, France (email: hamdan@ccr.jussieu.fr)
J.-P. THOUVENOT
Affiliation:
Laboratoire de Probabilités et Modèles aléatoires, UMR 7599, Universités Paris 6 et Paris 7, Boǐte Courrier 188, 4 Place Jussieu, 75252 Paris cedex 05, France (email: hamdan@ccr.jussieu.fr)
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Abstract

We consider stationary ergodic processes indexed by or whose finite-dimensional marginals have laws which are absolutely continuous with respect to Lebesgue measure. We define an entropy theory for these continuous processes, prove an analogue of the Shannon–MacMillan–Breiman theorem and study more precisely the particular example of Gaussian processes.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 2: William Parry Memorial Volume , April 2008 , pp. 447 - 480
Copyright
Copyright © Cambridge University Press 2008

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