Limit theorems for discrete-time markov chains on the nonnegative integers conditioned on recurrence to zero

Abstract

We consider a discrete–time Markov chain X on the state space with stationary one-step transition probabilities such that X is irreducible, transient, aperiodic and skip-free to the left. With denoting the modified Markov chain in which the states are aggregated into a single absorbing state, we study the conditional state probabilities of at time n, given that state 0 will be reached some time after time n. A sufficient condition for the convergence, as of these conditional probabilities to a proper distribution is determined, as well as a condition under which the limiting conditional distribution of X is the limit, as ,of the limiting conditional distribution of . For skip-free Markov chains we derive a necessary and sufficient condition for the existence of the limiting conditional distribution. As an example of a phenomenon which may be modelled by a limiting conditional distribution, we consider the backlog of a slotted ALOHA protocol