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.