On the Ergodicity of Certain Markov Chains in Random Environments

Balázs Gerencsér () and Miklós Rásonyi
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Balázs Gerencsér: Alfréd Rényi Institute of Mathematics and Eötvös Loránd University
Miklós Rásonyi: Alfréd Rényi Institute of Mathematics and Eötvös Loránd University

Journal of Theoretical Probability, 2023, vol. 36, issue 4, 2093-2125

Abstract: 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 $$X_t$$ X t are shown to converge to a limiting law in (weighted) total variation distance as $$t\rightarrow \infty $$ t → ∞ . 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.

Keywords: Ergodicity; Markov chain; Random environment; 60J05; 60K37; 37A25 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s10959-023-01256-7

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