Weak convergence of conditioned processes on a countable state space

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→∞. 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.