Quasi-Stationary Distributions and Convergence to Quasi-Stationarity of Birth-Death Processes

摘要

For a birth-death process (X(t), t = or > 0) on the state space (-1,0,1,...),where -1 is an absorbing state which is reached with certainty and (0,1,...) is an irreducible class, one addresses and solves three problems. First, one determines the set of initial distributions which are such that the distribution of X(t), conditioned on nonabsorption up to time t, is independent of t. Secondly, one determines the limiting conditional distribution of X(t), that is, the limit as t -> infinity of the distribution of X(t), conditioned on nonabsorption up to time t, for any initial distribution with finite support. Thirdly, one determines the rate of convergence of the distribution of X(t), conditioned on nonabsorption up to time t, to its limit. Some examples conclude the paper. The main tools are the spectral representation for the transition probabilities of the birth-death process and a duality concept for birth-death processes.