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

Abstract

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 < ∞) ≡ 1 and (*) lim i→∞ 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 ∊T ) < ∞ for some ∊ > 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.