Constructing Population Processes with Specified Quasi-Stationary Distributions

Abstract

This note is concerned with constructing time-homogeneous Markov chains on a countable state-space, including an absorbing state, such that the chain has a prescribed quasi-stationary distribution. The problem is characterized in terms of the underlying Q-matrix of the process. With no restriction on the Markov chain, it is straightforward to obtain a solution. For restricted processes this is no longer the case. We look in detail at population processes of birth–death and birth–catastrophe type, and in both cases obtain explicit constructions for Markov chains with any specified quasi-stationary distribution. The results are illustrated with examples.