Effective geometric ergodicity for Markov chains in random environment

Abstract In this short note we prove “effective” geometric ergodicity (i.e., a Perron–Frobenius theorem) for Markov chains in random mixing dynamical environment satisfying a random nonuniform version of the Doeblin condition. Effectivity here means that all the random variables involved in the random exponential rates are integrable with arbitrarily large order. This complements [Theorem 2.1 in Perron–Frobenius theorem, large deviations, and random perturbations in random environments, Math. Z. 1996], where “noneffective” geometric ergodicity was obtained. From a different perspective, our result is also motivated by ergodic theory, as it can be seen as an effective version of the “spectral” gap in the top Oseledets space in the Oseledets multiplicative ergodic theorem for the random Markov operator cocycle (when it applies). We also present applications of the effective ergodicity to rates in the (quenched) almost sure invariance principle (ASIP), exponential decay of correlations for Markovian skew products and for exponential tails bounds on random mixing times. As a byproduct of the proof of the ASIP rates we also provide easily verifiable sufficient conditions for checking the assumptions of [Theorem 2.1 in Limit theorems for random transformations and processes in random environments, Tran. Amer. Math. Soc. 1998].