Sharp Mixing Rates for Markov Chains on General Spaces with Unbounded Random Environments

TL;DR

This paper establishes sharp mixing rates and convergence properties for Markov chains in general spaces with unbounded random environments, extending existing results and achieving near-optimal rates for certain autoregressive processes.

Contribution

It introduces new convergence and mixing rate results for Markov chains in complex environments, significantly broadening prior theoretical understanding.

Findings

Proves convergence to a limiting law with explicit rates

Shows strong mixing properties and estimates mixing coefficients

Achieves near-optimal mixing rates for specific autoregressive models

Abstract

We consider Markov chains on general state spaces in stationary random environment which are defined by a random mapping that is contractive up to a bounded perturbation. We prove their convergence to a limiting law, providing convergence rates. We also show that these processes are strongly mixing and estimate their mixing coefficients. Our results significantly extend those available in the literature. In particular, for some additive autoregressive processes with exogenous covariates we achieve mixing rates that are optimal up to logarithmic factors.