  In this paper we consider the statistics of repeated measurements on the
output of a quantum Markov chain. We establish a large deviations result
analogous to Sanov's theorem for the empirical measure associated to finite
sequences of consecutive outcomes of a classical stochastic process. Our result
relies on the construction of an extended quantum transition operator (which
keeps track of previous outcomes) in terms of which we compute moment
generating functions, and whose spectral radius is related to the large
deviations rate function. As a corollary to this we obtain a central limit
theorem for the empirical measure. Such higher level statistics may be used to
uncover critical behaviour such as dynamical phase transitions, which are not
captured by lower level statistics such as the sample mean. As a step in this
direction we give an example of a finite system whose level-one rate function
is independent of a model parameter while the level-two rate is not.
