  We establish a central limit theorem for (a sequence of) multivariate
martingales which dimension potentially grows with the length $n$ of the
martingale. A consequence of the results are Gaussian couplings and a
multiplier bootstrap for the maximum of a multivariate martingale whose
dimensionality $d$ can be as large as $e^{n^c}$ for some $c>0$. We also develop
new anti-concentration bounds for the maximum component of a high-dimensional
Gaussian vector, which we believe is of independent interest.
  The results are applicable to a variety of settings. We fully develop its use
to the estimation of context tree models (or variable length Markov chains) for
discrete stationary time series. Specifically, we provide a bootstrap-based
rule to tune several regularization parameters in a theoretically valid
Lepski-type method. Such bootstrap-based approach accounts for the correlation
structure and leads to potentially smaller penalty choices, which in turn
improve the estimation of the transition probabilities.
