This note proves a uniform large deviation property for the empirical process of certain Markov chains that take values in a Polish space. The proof is based on recent results for the empirical measure.
Richard S. Ellis. Aaron D. Wyner. "Uniform Large Deviation Property of the Empirical Process of a Markov Chain." Ann. Probab. 17 (3) 1147 - 1151, July, 1989. https://doi.org/10.1214/aop/1176991261