  We study a variable length Markov chain model associated with a group of
stationary processes that share the same context tree but each process has
potentially different conditional probabilities. We propose a new model
selection and estimation method which is computationally efficient. We develop
oracle and adaptivity inequalities, as well as model selection properties, that
hold under continuity of the transition probabilities and polynomial
$\beta$-mixing. In particular, model misspecification is allowed.
  These results are applied to interesting families of processes. For Markov
processes, we obtain uniform rate of convergence for the estimation error of
transition probabilities as well as perfect model selection results. For chains
of infinite order with complete connections, we obtain explicit uniform rates
of convergence on the estimation of conditional probabilities, which have an
explicit dependence on the processes' continuity rates. Similar guarantees are
also derived for renewal processes.
  Our results are shown to be applicable to discrete stochastic dynamic
programming problems and to dynamic discrete choice models. We also apply our
estimator to a linguistic study, based on recent work, by Galves et al (2012),
of the rhythmic differences between Brazilian and European Portuguese.
