A universal large deviation lower bound is proved for sums of Banach space valued functions of an irreducible, general state space Markov chain. There are no restrictions on the functions (other than measurability).
"Large deviation lower bounds for arbitrary additive functionals of a Markov chain." Ann. Probab. 26 (4) 1660 - 1682, October 1998. https://doi.org/10.1214/aop/1022855877