Geometric Ergodicity and R-positivity for General Markov Chains

Abstract

We show that for positive recurrent Markov chains on a general state space, a geometric rate of convergence to the stationary distribution 7T in a "small" region ensures the existence of a uniform rate p < 1 such that for 7r-a.a. x, IlPn(x, .)-r(.)[l = O(pn). In particular, if there is a point a in the space with 2r(a) > 0, the result holds if IPn(a, a)-7r(a)l = O(p') for some pa < 1. This extends and strengthens the known results on a count-able 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 {RPn(x, A)}, as well as exhibiting the solidarity of ageometric rate of convergence for such sequences. We conclude by applying our results to random walk on a half-line.