On the ergodicity of certain Markov chains in random environments

Gerencsér, Balázs and Rásonyi, Miklós (2018) On the ergodicity of certain Markov chains in random environments. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. pp. 1-22. ISSN 0002-9947

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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 Xt are shown to converge to a limiting law in (weighted) total variation distance as t→∞. Convergence speed is estimated and an ergodic theorem is established for functionals of X. Our hypotheses on X combine the standard "small set" and "drift" 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 untractable. In particular, our results are pertinent to difference equations modulated by a stationary Gaussian process. Such equations arise in applications, for example, in discretized stochastic volatility models of mathematical finance.

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