Let {Xn, n = 0, 1, 2, ...} denote a Markov chain on a general state space and let f be a nonnegative function. The purpose of this paper is to present conditions which will imply that f(Xn) tends to 0 a.s., as n tends to infinity. As an application we obtain a result on synchronisation for random dynamical systems. At the end of the paper, we also present a result on convergence in distribution for Markov chains on general state spaces, thereby generalising a similar result for Markov chains on compact metric spaces.
AMS 2010 Subject Classification: 60J05, 60F15, 60F05.