Quasi-stationary distributions for birth-death processes with killing

Pauline Coolen-Schrijner and Erik A. van Doorn

International Journal of Stochastic Analysis, 2006, vol. 2006, 1-15

Abstract:

The Karlin-McGregor representation for the transition probabilities of a birth-death process with an absorbing bottom state involves a sequence of orthogonal polynomials and the corresponding measure. This representation can be generalized to a setting in which a transition to the absorbing state ( killing ) is possible from any state rather than just one state. The purpose of this paper is to investigate to what extent properties of birth-death processes, in particular with regard to the existence of quasi-stationary distributions, remain valid in the generalized setting. It turns out that the elegant structure of the theory of quasi-stationarity for birth-death processes remains largely intact as long as killing is possible from only finitely many states. In particular, the existence of a quasi-stationary distribution is ensured in this case if absorption is certain and the state probabilities tend to zero exponentially fast.

Date: 2006
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:084640

DOI: 10.1155/JAMSA/2006/84640

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