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Branching processes in random environments with thresholds

Part of: Markov processes Stochastic processes Limit theorems

Published online by Cambridge University Press:  22 August 2023

Giacomo Francisci Open the ORCID record for Giacomo Francisci [Opens in a new window]  and
Anand N. Vidyashankar
Giacomo Francisci*
Affiliation:
George Mason University
Anand N. Vidyashankar*
Affiliation:
George Mason University
*
*Postal address: Department of Statistics, George Mason University, 4000 University Drive, Fairfax, VA 22030, USA.
*Postal address: Department of Statistics, George Mason University, 4000 University Drive, Fairfax, VA 22030, USA.
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Abstract

Motivated by applications to COVID dynamics, we describe a model of a branching process in a random environment $\{Z_n\}$ 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, 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 distributions 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.

Keywords

BPRECOVID dynamicsergodicity of Markov chainsestimators of growth ratelength of cyclesmartingalesrandom sumsregenerative structuresize-dependent branching processsize-dependent branching process with a thresholdsubcritical regimesupercritical regime

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F05: Central limit and other weak theorems 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 92D25: Population dynamics (general) 92D30: Epidemiology 60G50: Sums of independent random variables; random walks
Type
Original Article
Information
Advances in Applied Probability , First View , pp. 1 - 50
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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