Quasi-stationary distributions for birth-death processes with killing

[摘要] The Karlin-McGregor representation for the transition probabilities of abirth-death process with an absorbing bottom state involves a sequence oforthogonal polynomials and the corresponding measure. This representation can begeneralized to a setting in which a transition to the absorbing state (killing) is possible from any state rather than just one state. The purpose ofthis 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 structureof the theory of quasi-stationarity for birth-death processes remains largelyintact as long as killing is possible from only finitely many states. Inparticular, the existence of a quasi-stationary distribution is ensured in thiscase if absorption is certain and the state probabilities tend to zeroexponentially fast.

[学科分类] 应用数学