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 $\pi$ in a "small" region ensures the existence of a uniform rate $\rho < 1$ such that for $\pi-\mathrm{a.a.} x, \|P^n(x, \bullet) - \pi(\bullet)\| = O(\rho^n)$. In particular, if there is a point $\alpha$ in the space with $\pi(\alpha) > 0$, the result holds if $|P^n(\alpha, \alpha) - \pi(\alpha)| = O(\rho^n_\alpha)$ for some $\rho_\alpha < 1$. 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 $\{R^nP^n(x, A)\}$, 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.