Ergodic theorems for Markov chains

In 1978 Atreia and Ney (Trans. Amer., Sos., 245, 493-501) and Nummelin (Z. Wahrscheinlichkeitstheorie verw. Gebiet, 43: 4, 309-318.) For the proof of ergodic theorems for recurrent Markov chains they proposed the so-called splitting method, which allows us to single out an atom in the extended phase space. In addition to the recurrence condition, one more condition is imposed: there is a set A of states and a nonnegative measure m, given on A, such that m minorizes the transition function or some its iteration on A. I must note that this condition was used to prove ergodic theorems in my article of 1965 (Sib. Mat., 6: 2, 413-432). The talk presents the method alternative to the method of splitting. In contrast to the probabilistic approach of the authors cited above, the method proposed by us is purely analytic. The starting point is the algebraic formula valid for elements of any ring. This formula made it possible to obtain a representation for the generating function of iterations of the transition function in the form of a fraction the numerator of which is an operator function analytic in the unit disk and the denominator is a scalar analytic function. The formula for the invariant measure and a series of identities connecting the various parameters of the chain are given as well. It should be noted that the algebraic formula mentioned above has a much wider scope than the case considered in the report.