6533b851fe1ef96bd12a8e24

RESEARCH PRODUCT

Context Trees, Variable Length Markov Chains and Dynamical Sources

Nicolas PouyannePeggy CénacFrédéric PaccautBrigitte Chauvin

subject

Discrete mathematicsPure mathematicsStationary distributionMarkov chain010102 general mathematicsProbabilistic dynamical sourcesProbabilistic logicContext (language use)Information theoryVariable length Markov chains01 natural sciencesMeasure (mathematics)Occurrences of words[MATH.MATH-PR]Mathematics [math]/Probability [math.PR]010104 statistics & probabilitysymbols.namesakesymbolsUniquenessDynamical systems of the intervalDirichlet series0101 mathematics[ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR]Dirichlet seriesMathematics

description

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.

yearjournalcountryeditionlanguage
2012-01-01
https://doi.org/10.1007/978-3-642-27461-9_1