Geometric Ergodicity and R-positivity for General Markov Chains

We show that for positive recurrent Markov chains on a general state space, a geometric rate of convergence to the stationary distribution in a "small" region ensures the existence of a uniform rate 0, the result holds if |Pⁿ (, ) - () | = O (ⁿ_) for some _ < 1. This extends and strengthens the known results on a countable state space. Our results are put in the more general R-theoretic context, and the methods we use enable us to establish the existence of limits for sequences \RⁿPⁿ (x, A) \, as well as exhibiting the solidarity of a geometric rate of convergence for such sequences. We conclude by applying our results to random walk on a half-line.