Limit theorems for inhomogeneous $\phi$-mixing Markov chains

Abstract

We prove limit theorems for inhomogeneous $\phi$-mixing Markov chains. In the case of uniform $\phi$-mixing, we will focus on the purely sequential (inhomogeneous) setting, while in the non-uniform case we will focus on Markov chains in random dynamical environments. A large part of the paper is devoted to verifying our main results for Markov chains satisfying a lower Doeblin condition and for Markov chains satisfying a Dobrushin-type contraction condition. We then apply our results in the random environment case to prove estimates on random mixing times as well as limit theorems for Markovian skew products. In fact, we show that one can estimate the $\alpha$-mixing coefficients of the coordinates of the skew product, which opens the door to a variety of limit theorems. Compared with \cite{dolgopyat2023berry}, we are able to prove optimal CLT rates for inhomogeneous Markov chains solely under $\phi$-mixing, without any ellipticity assumptions. Compared with existing results for Markov chains in random ergodic environments, under mixing conditions on the environment, we are able to treat non-uniformly $\phi$-mixing chains, including non-uniform Doeblin/Dobrushin conditions, while all existing results either hold for uniformly mixing chains or are formulated under hard-to-verify conditions.