Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes

For a birth-death process (X(t), t >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, we address and solve three problems. First, we determine the set of quasi-stationary distributions of the process, that is, the set of initial distributions which are such that the distribution of X(t), conditioned on non-absorption up to time t, is independent of t. Secondly, we determine the quasi-limiting distribution of X(t), that is, the limit as t--oo of the distribution of X(t), conditioned on non-absorption up to time t, for any initial distribution with finite support. Thirdly, we determine the rate of convergence of the transition probabilities of X(t), conditioned on non-absorption up to time t, to their limits. Some examples conclude the paper. Our main tools are the spectral representation for the transition probabilities of a birth-death process and a duality concept for birth-death processes.

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