The central limit theorem for Markov chains with general state space

We consider a Markov chain with general state space and an embedded Markov chain sampled at the times of successive returns to a subset $A_0$ of the state space. We assume that the latter chain is uniformly ergodic but the original Markov chain need not possess this property. We develop a modification of the spectral method and utilize it in proving the central limit theorem for the Markov chain under consideration.