A criterion on asymptotic stability for partially equicontinuous Markov operators

In this paper, we prove a slight, but practically useful, generalisation of a criterion on asymptotic stability for Markov e-chains by T. Szarek, which is based on the so-called lower bound technique, developed by A. Lasota and J. York. Simultaneously, we present an alternative proof of this theorem using an asymptotic coupling method introduced by M. Hairer. Our main result is illustrated by an application to random iterated function systems, which are not contracting on average.