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Weak convergence of conditioned processes on a countable state space. (English) Zbl 0839.60069
Summary: We consider the problem of conditioning a continuous-time Markov chain (on a countably infinite state space) not to hit an absorbing barrier before time $$T$$; and the weak convergence of this conditional process as $$T \to \infty$$. We prove a characterization of convergence in terms of the distribution of the process at some arbitrary positive time, $$t$$, introduce a decay parameter for the time to absorption, give an example where weak convergence fails, and give sufficient conditions for weak convergence in terms of the existence of a quasi-stationary limit, and a recurrence property of the original process.

##### MSC:
 60J27 Continuous-time Markov processes on discrete state spaces 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60B10 Convergence of probability measures
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