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Limiting Conditional Distributions for Birth-Death Processes

M. Kijima, M. G. Nair, P. K. Pollett and E. A. Van Doorn
Advances in Applied Probability
Vol. 29, No. 1 (Mar., 1997), pp. 185-204
DOI: 10.2307/1427866
Stable URL: http://www.jstor.org/stable/1427866
Page Count: 20
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Limiting Conditional Distributions for Birth-Death Processes
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Abstract

In a recent paper [16], one of us identified all of the quasi-stationary distributions for a non-explosive, 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.

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