  We show that large-scale typicality of Markov sample paths implies that the
likelihood ratio statistic satisfies a law of iterated logarithm uniformly to
the same scale. As a consequence, the penalized likelihood Markov order
estimator is strongly consistent for penalties growing as slowly as log log n
when an upper bound is imposed on the order which may grow as rapidly as log n.
Our method of proof, using techniques from empirical process theory, does not
rely on the explicit expression for the maximum likelihood estimator in the
Markov case and could therefore be applicable in other settings.
