RATIO LIMITS AND LIMITING CONDITIONAL DISTRIBUTIONS

FOR DISCRETE-TIME BIRTH-DEATH PROCESSES

ERIK A. VAN DOORN AND PAULINE SCHRIJNER

Faculty of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands

Submittend by Sandy L. Zabell

Received April 27, 1993

Abstract

We consider discrete-time birth-death processes with an absorbing state and study the conditional state distribution at time n given that absorption has not occurred by that time but will occur eventually. In particular, we establish conditions for the convergence of these distributions to a proper distribution as n → ∞. The problem turns out to be closely related to that of finding conditions for the existence of limits of ratios of n-step transition probabilities as n → ∞. Orthogonal polynomials feature in the spectral representation for the n-step transition probabilities of a birth-death process and, consequently, play a key role in the analysis.