Uniform CLT for Markov chains with a countable state space

For an irreducible Markov chain taking integer values which has a finite second moment for the first return time from some integer to itself, a necessary and sufficient condition for a CLT is found, which is uniform over all subsets of the integers. This is a generalization of one of the results in Durst and Dudley who worked in the i.i.d. case. An example that clarifies the difference between the i.i.d. case and Markov case is presented.