Multiplicative Ergodicity and Large Deviations for an Irreducible Markov Chain∗

The paper examines multiplicative ergodic theorems and the related multiplicative Poisson equation for an irreducible Markov chain on a countable state space. The partial products are considered for a realvalued function on the state space. If the function of interest satisfies a monotone condition, or is dominated by such a function, then (i) The mean normalized products converge geometrically quickly to a finite limiting value. (ii) The multiplicative Poisson equation admits a solution. (iii) Large deviation bounds are obtained for the empirical measures.