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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: [email protected])
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: [email protected])
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: [email protected])
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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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References

[1]Barron, A. R.. The strong ergodic theorem for densities: generalized Shannon–McMillan–Breiman theorem. Ann. Probab. 13(4) (1985), 1292–1303.CrossRefGoogle Scholar
[2]Doob, J. L.. Stochastic Processes. Wiley, New York, 1953.Google Scholar
[3]Dye, H. A.. On groups of measure preserving transformations. I. Amer. J. Math. 81 (1959), 119–159.CrossRefGoogle Scholar
[4]Helson, H.. Harmonic Analysis. Addison-Wesley, Reading, MA, 1983.Google Scholar
[5]Hoffman, K.. Banach Spaces of Analytic Functions. Prentice-Hall, Englewood Cliffs, NJ, 1962.Google Scholar
[6]Hoffman, C. and Rudolph, D. J.. Uniform endomorphisms which are isomorphic to Bernoulli shift. Ann. Math. (2) 156(1) (2002), 79–101.CrossRefGoogle Scholar
[7]Kolmogorov, A. N.. Stationary sequences in Hilbert space. Bull. Math. Univ. Moscow 2(6) (1941), 1–4.Google Scholar
[8]Mattila, P.. Geometry of Sets and Measures in Euclidean Spaces, Fractals and Rectifiability (Cambridge Studies in Advanced Mathematics, 44). Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[9]Ornstein, D. S.. Ergodic Theory, Randomness, and Dynamical Systems. Yale University Press, New Haven, 1974.Google Scholar
[10]Ornstein, D. S. and Weiss, B.. The Shannon–Mc Millan–Briman theorem for a class of amenable groups. Israel J. Math. 44(3) (1983), 53–60.CrossRefGoogle Scholar
[11]Parry, W.. Entropy and Generators in Ergodic Theory. W. A. Benjamin, New York, 1969.Google Scholar
[12]Parry, W.. Topics in Ergodic Theory. Cambridge University Press, Cambridge, 1981.Google Scholar
[13]Parry, W.. Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys. 16 (1996), 519–529.CrossRefGoogle Scholar
[14]Pinsker, M. S.. Information and Information Stability of Random Variables and Processes. Holden-Day, San Francisco, 1964.Google Scholar
[15]Rosenblatt, M.. Stationary processes as shifts of functions of independent random variables. J. Math. Mech. 8 (1959), 665–681.Google Scholar
[16]Shiriayev, A. N.. Probability, 2nd edn. Springer, Berlin, 1989.Google Scholar
[17]Shannon, C. E.. The mathematical theory of communication. Bell System Tech. J. 27 (1948), 379–423; 27 (1948), 623–656CrossRefGoogle Scholar
[18]Simon, B.. The sharp form of the strong Szegö theorem. Geometry, Spectral Theory, Groups, and Dynamics (Contemporary Mathematics, 3877). American Mathematical Society, Providence, RI, 2005, pp. 253–275.CrossRefGoogle Scholar
[19]Smorodinsky, M.. Ergodic Theory, Entropy (Lecture Notes in Mathematics, 214). Springer, Berlin, 1971.CrossRefGoogle Scholar
[20]Wiener, N.. Extrapolation, Interpolation and Smoothing of Stationary Time Series. Wiley, New York, 1945.Google Scholar
[21]Wiener, N.. Non-Linear Problems in Random Theory. MIT Press, Cambridge, MA; Wiley, New York, 1958.Google Scholar