Exponential inequalities for VLMC empirical trees

¹ Instituto de Matemática e Estatística, Universidade de São Paulo, BP 66281, 05315-970 São Paulo, Brasil; galves@ime.usp.br
² Institut de Mathématiques de Bourgogne, BP 47870, 21078 Dijon cedex France; vmaume@u-bourgogne.fr; schmittb@u-bourgogne.fr

Received: 22 October 2006
Revised: 10 June 2007

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.