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Branching processes in generalized autoregressive conditional environments

Part of: Markov processes Stochastic processes Limit theorems Inference from stochastic processes

Published online by Cambridge University Press:  11 January 2017

Irene Hueter
Irene Hueter*
Affiliation:
Columbia University
*
* Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, USA. Email address: irene.hueter@columbia.edu
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Abstract

Branching processes in random environments have been widely studied and applied to population growth systems to model the spread of epidemics, infectious diseases, cancerous tumor growth, and social network traffic. However, Ebola virus, tuberculosis infections, and avian flu grow or change at rates that vary with time—at peak rates during pandemic time periods, while at low rates when near extinction. The branching processes in generalized autoregressive conditional environments we propose provide a novel approach to branching processes that allows for such time-varying random environments and instances of peak growth and near extinction-type rates. Offspring distributions we consider to illustrate the model include the generalized Poisson, binomial, and negative binomial integer-valued GARCH models. We establish conditions on the environmental process that guarantee stationarity and ergodicity of the mean offspring number and environmental processes and provide equations from which their variances, autocorrelation, and cross-correlation functions can be deduced. Furthermore, we present results on fundamental questions of importance to these processes—the survival-extinction dichotomy, growth behavior, necessary and sufficient conditions for noncertain extinction, characterization of the phase transition between the subcritical and supercritical regimes, and survival behavior in each phase and at criticality.

Keywords

Branching processes in random environmentGARCHGalton–Watson processextinctionphase transitionlimit theorems

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 62M10: Time series, auto-correlation, regression, etc. 60G10: Stationary processes 60F05: Central limit and other weak theorems
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 4 , December 2016 , pp. 1211 - 1234
Copyright
Copyright © Applied Probability Trust 2017 

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