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 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.

Markov chains in random environment with applications in queueing theory and machine learning

“…In contrast with the drift condition used in [8] (cf. Assumption 2.2 on page 2), the domain of γ is and not .…”

“…The article [8], introducing new tools, managed to establish the existence of limiting laws and ergodic theorems for certain classes of MCREs which satisfy suitable versions of the standard drift and minorization conditions of Markov chain theory (as presented e.g. in [14]).…”

“…Assumption 2.2 of [8], however, severely restricted the scope of applications by requiring that the system dynamics is contractive whatever the state of the random environment is. The present study aims to remove this restriction: we require only that process dynamics is contractive on the average, in the sense of Assumption 2.3 below.…”

Invariant measures for multidimensional fractional stochastic volatility models

“…We have verified the minorization Assumption 2.9 just before so Theorem 2.11 applies, ensuring convergence in total variation. The present example complements Example 3.4 of [12] where convergence in total variation was established under stronger assumptions (but with a rate estimate).…”

“…First, we point out that ( 16) could be established for certain non-Markovian models like (25) below (which are not covered by current literature). Second, using technology from [12,21], various mixing properties and laws of large numbers (with rate estimates) could be established for functionals of the process X t , t ∈ N. Third, central limit theorems can be derived from mixing conditions, just like in [32].…”