Slow recurrent regimes for a class of one-dimensional stochastic growth models

TL;DR

This paper classifies behaviors of one-dimensional stochastic growth models, providing bounds on hitting times and conditions for recurrence, with applications to Galton-Watson processes and extinction times.

Contribution

It offers nearly optimal bounds for hitting times and characterizes recurrence types in a broad class of stochastic growth models, including state-dependent Galton-Watson processes.

Findings

Bounds for tail of hitting times of compact sets

Criteria for null or positive recurrence in Markov chains

Subgeometric convergence to invariant measures

Abstract

We classify the possible behaviors of a class of one-dimensional stochastic recurrent growth models. In our main result, we obtain nearly optimal bounds for the tail of hitting times of some compact sets. If the process is an aperiodic irreducible Markov chain, we determine whether it is null recurrent or positive recurrent and in the latter case, we obtain a subgeometric convergence of its transition kernel to its invariant measure. We apply our results in particular to state-dependent Galton-Watson processes and we give precise estimates of the tail of the extinction time.