Context Trees, Variable Length Markov Chains and Dynamical Sources

  • Peggy Cénac
  • Brigitte Chauvin
  • Frédéric Paccaut
  • Nicolas Pouyanne
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2046)

Abstract

Infinite random sequences of letters can be viewed as stochastic chains or as strings produced by a source, in the sense of information theory. The relationship between Variable Length Markov Chains (VLMC) and probabilistic dynamical sources is studied. We establish a probabilistic frame for context trees and VLMC and we prove that any VLMC is a dynamical source for which we explicitly build the mapping. On two examples, the “comb” and the “bamboo blossom”, we find a necessary and sufficient condition for the existence and the uniqueness of a stationary probability measure for the VLMC. These two examples are detailed in order to provide the associated Dirichlet series as well as the generating functions of word occurrences.

Keywords

Variable length Markov chains Dynamical systems of the interval Dirichlet series Occurrences of words Probabilistic dynamical sources 

Notes

Acknowledgements

We are very grateful to Antonio Galves, who introduced us to the challenging VLMC topics. We warmly thank Brigitte Vallée for valuable and stormy discussions.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Peggy Cénac
    • 1
  • Brigitte Chauvin
    • 2
    • 3
  • Frédéric Paccaut
    • 4
  • Nicolas Pouyanne
    • 3
  1. 1.Institut de Mathématiques de Bourgogne IMB UMR 5584 CNRSUniversité de BourgogneDIJON CEDEX, BourgogneFrance
  2. 2.INRIA Rocquencourtproject AlgorithmsLe Chesnay CEDEXFrance
  3. 3.Laboratoire de Mathématiques de Versailles, CNRS, UMR 8100Université de Versailles - St-QuentinVersailles CEDEXFrance
  4. 4.LAMFA, CNRS, UMR 6140Université de Picardie Jules VerneAmiensFrance

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