This book studies the large deviations for empirical measures and
vector-valued additive functionals of Markov chains with general
state space. Under suitable recurrence conditions, the ergodic
theorem for additive functionals of a Markov chain asserts the
almost sure convergence of the averages of a real or vector-valued
function of the chain to the mean of the function with respect to
the invariant distribution. In the case of empirical measures, the
ergodic theorem states the almost sure convergence in a suitable
sense to the invariant distribution. The large deviation theorems
provide precise asymptotic estimates at logarithmic level of the
probabilities of deviating from the preponderant behavior asserted
by the ergodic theorems.
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