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

TL;DR

This paper investigates a population model in a random environment, establishing conditions for null recurrence and transience, and revealing a unique phase transition behavior where the process shifts from transience to null recurrence without becoming positive recurrent.

Contribution

It provides new sufficient conditions for null recurrence and transience in a binomial catastrophe model with random environment, including a novel phase transition phenomenon.

Findings

Conditions for null recurrence and transience are derived.

A specific case shows a phase transition from transience to null recurrence.

The model exhibits a unique transition without positive recurrence.

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 $\beta_n$. The sequence $(\beta_n)$ 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 $\beta$. This particular case shows a rather unusual phase transition in $\beta$ in the sense that the Markov chain goes from transience to null recurrence without ever reaching positive recurrence.