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Examples of convergence and non-convergence of Markov chains conditioned not to die. (English) Zbl 1014.60074

Authors’ abstract: The authors give two examples of evanescent Markov chains which exhibit unusual behavior on conditioning to survive for large times. In the first example they show that the conditioned processes converge vaguely in the discrete topology to a limit with a finite lifetime, but converge weakly in the Martin topology to a non-Markovian limit. In the second example, although the family of conditioned laws are tight in the Martin topology, they possess multiple limit points so that weak convergence fails altogether.
Reviewer: Uwe Rösler (Kiel)

MSC:

60J50 Boundary theory for Markov processes
60B10 Convergence of probability measures
60G99 Stochastic processes
60J27 Continuous-time Markov processes on discrete state spaces
60J45 Probabilistic potential theory
60J55 Local time and additive functionals
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