The ergodic behaviour of random walks

1. Many stochastic processes occurring in practice may be formulated as Markov chains with an enumerable state space. It is then important to know whether or not the chain is ergodic, i.e. whether or not a stationary distribution exists. For particular problems this has often been determined by complex and ingenious methods, a good example being the analysis by Kiefer & Wolfowitz (1955) of the many-server queue. However, Foster (1953) has given a general criterion for a chain to be ergodic, and the purpose of this paper is to examine the way in which his result may be applied to particullar processes. By way of example, the technique is applied to two important problems in queueing theory. The results obtained also have consequences in the theory of multi-dimensional random walks, in which context ergodicity simply means that the particle does not escape to infinity. The processes with which we will be concerned are those which may be formulated as Markov chains on the state space of vectors