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Exponential inequalities for VLMC empirical trees

Published online by Cambridge University Press:  23 January 2008

Antonio Galves ,
Véronique Maume-Deschamps  and
Bernard Schmitt
Antonio Galves
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, BP 66281, 05315-970 São Paulo, Brasil; galves@ime.usp.br
Véronique Maume-Deschamps
Affiliation:
Institut de Mathématiques de Bourgogne, BP 47870, 21078 Dijon cedex France; vmaume@u-bourgogne.fr; schmittb@u-bourgogne.fr
Bernard Schmitt
Affiliation:
Institut de Mathématiques de Bourgogne, BP 47870, 21078 Dijon cedex France; vmaume@u-bourgogne.fr; schmittb@u-bourgogne.fr
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Abstract

A seminal paper by Rissanen, published in 1983, introduced the class of Variable Length Markov Chains and the algorithm Context which estimates the probabilistic tree generating the chain. Even if the subject was recently considered in several papers, the central question of the rate of convergence of the algorithm remained open. This is the question we address here. We provide an exponential upper bound for the probability of incorrect estimation of the probabilistic tree, as a function of the size of the sample. As a consequence we prove the almost sure consistency of the algorithm Context. We also derive exponential upper bounds for type I errors and for the probability of underestimation of the context tree. The constants appearing in the bounds are all explicit and obtained in a constructive way.

Keywords

Variable Length Markov Chaincontext treealgorithm contextweak dependance
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 12 , 2008 , pp. 219 - 229
Copyright
© EDP Sciences, SMAI, 2008

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