Weak convergence of conditioned birth-death processes in discrete time

Abstract

We consider a discrete-time birth-death process on the non-negative integers with −1 as an absorbing state and study the limiting behaviour as n → ∞ of the process conditioned on non-absorption until time n. By proving that a condition recently proposed by Martinez and Vares is vacuously true, we establish that the conditioned process is always weakly convergent when all self-transition probabilities are zero. In the aperiodic case we obtain a necessary and sufficient condition for weak convergence.