Null recurrence and transience for a binomial catastrophe model in random environment

Abstract

We consider a discrete time population model for which each individual alive at time n survives independently of everybody else at time n + 1 with probability β n . The sequence ( β n ) is i.i.d. and constitutes our random environment. Moreover, at every time n we add Z n individuals to the population. The sequence ( Z n ) is also i.i.d. We find sufficient conditions for null recurrence and transience (positive recurrence has been addressed by Neuts 1994 J. Appl. Probab. 31 48–58). We apply our results to a particular ( Z n ) distribution and deterministic β. This particular case shows a rather unusual phase transition in β in the sense that the Markov chain goes from transience to null recurrence without ever reaching positive recurrence.