A simple lemma concerning the Doeblin minorization condition and its applications to limit theorems for inhomogeneous Markov chains

Abstract

In this short note we provide an elementary proof that a certain type of nonuniform sequential Doeblin minorization condition implies non-uniform sequential "geometric" ergodicity. Using this result several limit theorems for inhomogeneous Markov chains follow immediately from existing results [7,8,12]. We then focus our attention to Markov chains in random dynamical environment and deduce effective mixing rates which imply limit theorems for such processes by using the methods of [16] (which were formulated in a dynamical setup). The crucial part of the proof is to obtain effective convergence rates towards the random equivariant distribution, which has its own interest and yields, for instance, effective mixing times estimates together with results for the corresponding skew products