The spectral method and the central limit theorem for general Markov chains

We consider Markov chains with an arbitrary phase space and develop a modification of the spectral method that enables us to prove the central limit theorem (CLT) for non-uniformly ergodic Markov chains. The conditions imposed on the transition function are more general than those by Athreya–Ney and Nummelin. Our proof of the CLT is purely analytical.