The CLT for Markov chains with a countable state space embedded in the space lp

We find necessary and sufficient conditions for the CLT for Markov chains with a countable state space embedded in the space lp for p[greater-or-equal, slanted]1. This result is an extension of the uniform CLT over the family of indicator functions in Levental (Stochastic Processes Appl. 34 (1990) 245-253), where the result is equivalent to our case p=1. A similar extension for the uniform CLT over a family of possibly unbounded functions in Tsai (Taiwan. J. Math. 1(4) (1997) 481-498) is also obtained.