Quasi-stationary laws for Markov processes: Examples of an always proximate absorbing state

Under consideration is a continuous-time Markov process with non-negative integer state space and a single absorbing state 0. Let T be the hitting time of zero and suppose P-i(T <x) = 1 and (*) lim(i-->x)P(i)(T > t) = 1 for all t > 0. Most known cases satisfy (*). The Markov process has a quasi-stationary distribution iff E(i)(e(epsilon T)) <infinity for some epsilon > 0.The published proof of this fact makes crucial use of (*). By means of examples it is shown that (*) can be violated in quite drastic ways without destroying the existence of a quasi-stationary distribution.