Recurrence criteria for multi-dimensional Markov chains and multi-dimensional linear birth and death processes

Abstract

Criteria are established for a discrete-time Markov process {Xn } n≧0 in R d to have strictly positive, respectively zero, probability of escaping to infinity. These criteria are mainly in terms of the mean displacement vectors μ(y) = E{X n+1|Xn = y} – y, and are essentially such that they force a deterministic process w.p.1 to move off to infinity, respectively to return to a compact set infinitely often. As an application we determine of most two-dimensional birth and death processes with rates linearly dependent on the population, whether they can escape to infinity or not.