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 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, Volume 12 (2008), pp. 219-229. doi : 10.1051/ps:2007035. http://www.numdam.org/articles/10.1051/ps:2007035/

[1] Variations on probabilistic suffix trees: statistical modeling and prediction of protein families. Bioinformatics 17 (2001) 23-43.

and ,[2] Variable length Markov chains. Ann. Statist. 27 (1999) 480-513. | MR | Zbl

and ,[3] Large-scale typicality of Markov sample paths and consistency of MDL order estimators. Special issue on Shannon theory: perspective, trends, and applications. IEEE Trans. Inform. Theory 48 (2002) 1616-1628. | MR | Zbl

,[4] Context tree estimation for not necessarily finite memory processes via BIC and MDL, manuscript (2005). | MR | Zbl

and ,[5] A new covariance inequality and applications. Stochastic Process. Appl. 106 (2003) 63-80. | MR | Zbl

and ,[6] New dependence coefficients. Examples and applications to statistics. Prob. Theory Relat. Fields 132 (2005) 203-236. | MR | Zbl

and ,[7] Coupling and regeneration for stochastic processes2000).

and ,[8] Estimation of general stationary processes by variable length Markov chains. Scand. J. Statist. 30 (2003) 459-480. | MR | Zbl

and ,[9] Sequence Motif identification and protein classification using probabilistic trees. Lect. Notes Comput. Sci. 3594 (2005) 190-193.

and ,[10] Exponential inequalities and estimation of conditional probabilities in Dependence in probability and statistics, Lect. Notes in Stat., Vol. 187, P. Bertail, P. Doukhan and P. Soulier Eds. Springer (2006). | MR

,[11] A universal data compression system. IEEE Trans. Inform. Theory 29 (1983) 656-664. | MR | Zbl

,[12] Implementing the context-tree weighting method: arithmetic coding. Recent advances in interdisciplinary mathematics (Portland, ME, 1997). J. Combin. Inform. System Sci. 25 (2000) 49-58. | MR

and ,[13] The context-tree weighting method: basic properties. IEEE Trans. Inform. Theory 41 (1995) 653-664. | Zbl

, and ,*Cited by Sources: *