Branching Processes in Random Environments with Thresholds

TL;DR

This paper models a COVID-inspired branching process with thresholds that cause cyclical behavior, identifying subsequences with Markov properties, and establishing limit theorems for regime durations and proportions.

Contribution

It introduces a novel threshold-based branching process model with regenerative subsequences, providing limit theorems and explicit variance formulas for regime durations.

Findings

Subsequences at crossing times are Markovian.

Regenerative structure of subsequences is established.

Limit theorems for regime durations and proportions are proved.

Abstract

Motivated by applications to COVID dynamics, we describe a branching process in random environments model $\{Z_n\}$ whose characteristics change when crossing upper and lower thresholds. This introduces a cyclical path behavior involving periods of increase and decrease leading to supercritical and subcritical regimes. Even though the process is not Markov, we identify subsequences at random time points $\{(\tau_j, \nu_j)\}$ - specifically the values of the process at crossing times, {\it{viz.}}, $\{(Z_{\tau_j}, Z_{\nu_j})\}$ - along which the process retains the Markov structure. Under mild moment and regularity conditions, we establish that the subsequences possess a regenerative structure and prove that the limiting normal distribution of the growth rates of the process in supercritical and subcritical regimes decouple. For this reason, we establish limit theorems concerning the length of supercritical and subcritical regimes and the proportion of time the process spends in these regimes. As a byproduct of our analysis, we explicitly identify the limiting variances in terms of the functionals of the offspring distribution, threshold distribution, and environmental sequences.