Limiting conditional distributions for birth-death processes

Masaaki Kijima, Gopal Nair, Phil Pollett and Erik van Doorn

Abstract: In a recent paper [1], one of us identified all of the quasi-stationary distributions for a regular, evanescent birth-death process for which absorption is certain, and established conditions for the existence of the corresponding limiting conditional distributions. Our purpose is to extend these results in a number of directions. We shall consider separately two cases depending on whether or not the process is evanescent. In the former case we shall relax the condition that absorption is certain. Furthermore, we shall allow for the possibility that the minimal process might be explosive, so that the transition rates alone will not necessarily determine the birth-death process uniquely. Although we shall be concerned mainly with the minimal process, our most general results hold for any birth-death process whose transition probabilities satisfy both the backward and the forward Kolmogorov differential equations.

1. van Doorn, E.A. (1991) Quasi-stationary distributions and convergence to quasi-stationarity of a birth-death process, Adv. Appl. Probab. 23 , 683--700.

AMS 1991 Subject Classification: 60J80; 60J27.

Keywords: Invariant measures; quasi-stationary distributions.

Acknowledgement: This worked was funded by the Australian Research Council and the University of Queensland.

The authors:

• Masaaki Kijima, Graduate School of Systems Management, University of Tsukuba. kijima@gssm.otsuka.tsukuba.ac.jp
• Gopal Nair, School of Mathematics and Statistics, Curtin University of Technology. gopal@cs.curtin.edu.au
• Phil Pollett, Discipline of Mathematics, The University of Queensland. pkp@maths.uq.edu.au
• Erik van Doorn, Faculty of Applied Mathematics, University of Twente. doorn@math.utwente.nl

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