WEAK CONVERGENCE OF CONDITIONED BIRTH-DEATH

PROCESSES IN DISCRETE TIME

PAULINE SCHRIJNER^{⋆}, University of Durham ERIK A. VAN DOORN^{⋆⋆}, University of Twente

Abstract

We consider a discrete-time birth-death process on the nonnegative integers with −1 as an absorbing state and study the limiting behaviour as n → ∞ of the process conditioned on nonabsorption until time n. By proving that a condition recently proposed by Martínez 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.

BIRTH-DEATH PROCESS, CONDITIONED PROCESS, RATIO LIMIT, WEAK CONVERGENCE AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60J80

⋆Postal address: Department of Mathematics, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, England. E-mail address: pauline.schrijner@durham.ac.uk ⋆⋆Postal address: Faculty of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. E-mail address: doorn@math.utwente.nl