LIMIT THEOREMS FOR DISCRETE-TIME MARKOV CHAINS

ON THE NONNEGATIVE INTEGERS CONDITIONED ON RECURRENCE TO ZERO

Erik A. van Doorn

Faculty of Applied Mathematics, University of Twente P.O. Box 217, 7500 AE Enschede, The Netherlands e-mail address: doorn@math.utwente.nl

Pauline Schrijner

Department of Mathematics and Statistics, University of Newcastle Newcastle upon Tyne NE1 7RU, UK

Abstract. We consider a discrete-time Markov chain X on the state space ℕ ≡ {0, 1, …} with stationary one-step transition probabilities such that X is irreducible, transient, aperiodic and skip-free to the left. With X^{(m)} denoting the modified Markov chain in which the states {m, m + 1, …} are aggregated into a single absorbing state, we study the conditional state probabilities of X (resp. X^{(m)}) at time n, given that state 0 will be reached some time after time n. A sufficient condition for the convergence, as n → ∞, 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 m → ∞, of the limiting conditional distribution of X^{(m)}. 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.

Keywords. geometric transience, limiting conditional distribution, R-invariant measure, R-invariant vector, R-recurrence, R-transience, skip-free Markov chains, slotted ALOHA.