The Analytical Approach to Recurrent Markov Chains Alternative to the Splitting Method and Its Applications

We consider the Harris Markov chain with the general phase space. It is supposed that the transition probability majorizes a nonnegative measure on some subset of a phase space for any initial state from this subset. Up to now such chains are studying via so-called splitting method introduced by Athreya-Ney and Nummelin. We suggest the analytical approach which allows to prove both ergodic theorems and CLT for sums of random variables defined on the Markov chain. We state also the probability inequality which sharpens that by Bertail and Clemencon.