On the Ergodicity of Certain Markov Chains in Random Environments

Abstract

We study the ergodic behaviour of a discrete-time process X which is a Markov chain in a stationary random environment. The laws of X_t are shown to converge to a limiting law in (weighted) total variation distance as trightarrow infty . Convergence speed is estimated, and an ergodic theorem is established for functionals of X. Our hypotheses on X combine the standard “drift” and “small set” conditions for geometrically ergodic Markov chains with conditions on the growth rate of a certain “maximal process” of the random environment. We are able to cover a wide range of models that have heretofore been intractable. In particular, our results are pertinent to difference equations modulated by a stationary (Gaussian) process. Such equations arise in applications such as discretized stochastic volatility models of mathematical finance.