Context trees, variable length Markov chains and dynamical sources.

Peggy Cénac (1) , Brigitte Chauvin (2, 3) , Frédéric Paccaut (4) , Nicolas Pouyanne (2)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 LMV - Laboratoire de Mathématiques de Versailles
3 ALGORITHMS - Algorithms
4 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352

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.

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

Domaines: Probabilités [math.PR]

Peggy Cénac, Brigitte Chauvin, Frédéric Paccaut, Nicolas Pouyanne. Context trees, variable length Markov chains and dynamical sources.. Catherine Donati-Martin and Antoine Lejay and Alain Rouault. Séminaire de Probabilités XLIV, Springer, pp.1-39, 2012, Lecture Notes in Mathematics, 978-3-642-27460-2. ⟨10.1007/978-3-642-27461-9_1⟩. ⟨hal-00794627⟩