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.
Mots-clés : variable length Markov chain, context tree, algorithm context, weak dependance
@article{PS_2008__12__219_0,
author = {Galves, Antonio and Maume-Deschamps, V\'eronique and Schmitt, Bernard},
title = {Exponential inequalities for {VLMC} empirical trees},
journal = {ESAIM: Probability and Statistics},
pages = {219--229},
publisher = {EDP-Sciences},
volume = {12},
year = {2008},
doi = {10.1051/ps:2007035},
mrnumber = {2374639},
zbl = {1182.62165},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2007035/}
}
TY - JOUR AU - Galves, Antonio AU - Maume-Deschamps, Véronique AU - Schmitt, Bernard TI - Exponential inequalities for VLMC empirical trees JO - ESAIM: Probability and Statistics PY - 2008 SP - 219 EP - 229 VL - 12 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2007035/ DO - 10.1051/ps:2007035 LA - en ID - PS_2008__12__219_0 ER -
%0 Journal Article %A Galves, Antonio %A Maume-Deschamps, Véronique %A Schmitt, Bernard %T Exponential inequalities for VLMC empirical trees %J ESAIM: Probability and Statistics %D 2008 %P 219-229 %V 12 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2007035/ %R 10.1051/ps:2007035 %G en %F PS_2008__12__219_0
Galves, Antonio; Maume-Deschamps, Véronique; Schmitt, Bernard. Exponential inequalities for VLMC empirical trees. ESAIM: Probability and Statistics, Tome 12 (2008), pp. 219-229. doi : 10.1051/ps:2007035. http://www.numdam.org/articles/10.1051/ps:2007035/
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