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 π in a "small" region ensures the existence of a uniform rate ρ < 1 such that for π-a.a. x, \|Pⁿ(x, •) - π(•)\| = O(ρⁿ). In particular, if there is a point α in the space with π(α) > 0, the result holds if |Pⁿ(α, α) - π(α)| = O(ρⁿ_α) for some ρ_α < 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 ⁿPⁿ(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.