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

2022/11/25 by Luiz Renato Fontes, Fontes, Luiz Renato, Fábio P. Machado +3
Mathematics · Physics and Astronomy · #Stochastic processes and statistical mechanics #Complex Network Analysis Techniques

paper · pdf · doi:10.48550/arxiv.2211.14193

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). 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.

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