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

Let {Xn} be a [empty set][combining character]-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 [epsilon] > 0 such that E {||Xn+1|| -- ||Xn|| | Xn = x} [less-than-or-equals, slant] -[epsilon] whenever x lies outside K and E{||Xn+1|| | Xn=x} is bounded, x [set membership, variant] K; whilst the conditions for recurrence will be met if there exists a compact K with E {||Xn+1|| - ||Xn|| | Xn = x} [less-than-or-equals, slant] 0 for all x outside K. An application to queueing theory is given.