Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space

Abstract

Let {Xn} be a ∅-irreducible Markov chain on an arbitrary space. Sufficient conditions are given under which the chain is ergodic or recurrent. These extend known results for chains on a countable state space. In particular, it is shown that if the space is a normed topological space, then under some continuity conditions on the transition probabilities of {Xn} the conditions for ergodicity will be met if there is a compact set K and an ϵ > 0 such that E {‖Xn+1‖ — ‖Xn‖ ∣ Xn = x} ⩽ −ϵ whenever x lies outside K and E{‖Xn+1‖ ∣ Xn=x} is bounded, x ∈ K; whilst the conditions for recurrence will be met if there exists a compact K with E {‖Xn+1‖ − ‖Xn‖ ∣ Xn = x} ⩽ 0 for all x outside K. An application to queueing theory is given.