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

Abstract

Let { X n } 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 { X n } the conditions for ergodicity will be met if there is a compact set K and an ϵ > 0 such that E {‖X n+1‖ — ‖X n‖ ∣ X n = x} ⩽ −ϵ whenever x lies outside K and E{‖X n+1‖ ∣ X n=x} is bounded, x ∈ K; whilst the conditions for recurrence will be met if there exists a compact K with E {‖X n+1‖ − ‖X n‖ ∣ X n = x} ⩽ 0 for all x outside K. An application to queueing theory is given.