TWO THEOREMS ON CONVERGENCE PARAMETER OF AN IRREDUCIBLE MARKOV CHAIN

A homogeneous irreducible Markov chain X with a measurable state space (E, B) and a transient operator P acting in a probability of bounded from below measurable functions is considered. The sigma-algebra beta is assumed to be countably generated. It is proved that if the chain is aperiodic and function f and measure. are small, then [nu(P(n)f)](1/n) -> R as n -> infinity, where R is the convergence parameter. For periodic Markov chains this statement can be modified in the following way. If a chain X is symmetric with respect to some sigma-finite measure pi, then R = vertical bar vertical bar(P) over tilde vertical bar vertical bar(-1), where (P) over tilde is a bounded self-adjoint operator generated by P and acting in the space L-2(pi). Results of this paper extend the results of M. G. Shur