Domain of Attraction of the Quasi-Stationary Distribution for the Linear Birth and Death Process with Killing

The model of linear birth and death processes with killing has been studied by Karlin and Tavar (1982). This paper is concerned with three problems in connection with quasi-stationary distributions (QSDs) for linear birth-death process with killing on a semi-infinite lattice of integers. The first problem is to determine the decay parameter of . We have where , , are the birth, death and killing rates in state , respectively. The second one is to prove the uniqueness of the QSD which is a geometric distribution. It is interesting to find that the unkilled process has a one-parameter family of QSDs while the killed process has precisely one QSD. The last one is to solve the domain of attraction problem, that is, we obtain that any initial distribution is in the domain of attraction of the unique QSD for . Our study is motivated by the population genetics problem.