Journal of Applied Mathematics and Stochastic Analysis
Volume 2006 (2006), Article ID 84640, 15 pages
doi:10.1155/JAMSA/2006/84640
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.