Quasi‐stationary distributions for birth‐death processes with killing

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.