The main theorems in this paper prove uniform large deviation properties for the empirical measure and the multivariate empirical measure of a Markov chain that takes values in a complete separable metric space. One contribution of the paper is that, in contrast to previous large deviation results for the empirical measure, we do not assume that the transition probability of the Markov chain has a density with respect to a reference measure.
Richard S. Ellis. "Large Deviations for the Empirical Measure of a Markov Chain with an Application to the Multivariate Empirical Measure." Ann. Probab. 16 (4) 1496 - 1508, October, 1988. https://doi.org/10.1214/aop/1176991580