EXISTENCE OF QUASI-STATIONARY DISTRIBUTIONS - A RENEWAL DYNAMICAL-APPROACH

We consider Markov processes on the positive integers for which the origin is an absorbing state. Quasi-stationary distributions (qsd's) are described as fixed points of a transformation Phi in the space of probability measures. Under the assumption that the absorption time at the origin, R, of the process starting from state x goes to infinity in probability as x --> infinity, Pie show that the existence of a qsd is equivalent to E(x)e(lambda R) < infinity for some positive lambda and x. We also prove that a subsequence of Phi(n) delta(x) converges to a minimal qsd. For a birth and death process we prove that Phi(n) delta(x) converges along the full sequence to the minimal qsd. The method is based on the study of the renewal process with interarrival times distributed as the absorption time of the Markov process with a given initial measure mu. The key tool is the fact that the residual time in that renewal process has as stationary distribution the distribution of the absorption time of Phi mu.