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.
AMS Classification: 60J05, 37E05
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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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Cénac, P., Chauvin, B., Paccaut, F., Pouyanne, N. (2012). Context Trees, Variable Length Markov Chains and Dynamical Sources. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIV. Lecture Notes in Mathematics(), vol 2046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27461-9_1
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DOI: https://doi.org/10.1007/978-3-642-27461-9_1
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