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May 2022 Defective Galton-Watson processes in a varying environment
Götz Kersting, Carmen Minuesa
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Bernoulli 28(2): 1408-1431 (May 2022). DOI: 10.3150/21-BEJ1393

Abstract

We study an extension of the so-called defective Galton-Watson processes obtained by allowing the offspring distribution to change over the generations. Thus, in these processes, the individuals reproduce independently of the others and in accordance to some possibly defective offspring distribution depending on the generation. Moreover, the defect 1fn(1) of the offspring distribution at generation n represents the probability that the process hits an absorbing state Δ at that generation. We focus on the asymptotic behaviour of these processes. We establish the almost sure convergence of the process to a random variable with values in N0{Δ} and we provide two characterisations of the duality extinction-absorption at Δ. We also state some results on the absorption time and the properties of the process conditioned upon its non-absorption, some of which require us to introduce the notion of defective branching trees in varying environment.

Funding Statement

Carmen Minuesa’s research has been supported by the Ministerio de Economía y Competitividad (grant MTM2015-70522-P), the Spanish State Research Agency (PID2019-108211GBI00/AEI/10.13039/ 501100011033), the Junta de Extremadura and the European Regional Development Fund (grants IB16099 and GR18103).

Acknowledgements

This research was initiated while Carmen Minuesa was visiting the Institute of Mathematics, Goethe University Frankfurt, in Frankfurt am Main, and she is grateful for the hospitality and collaboration.

The authors would like to thank Serik Sagitov (Chalmers University of Technology and University of Gothenburg) for suggesting the topic of this research. The authors also thank the anonymous referees for their valuable comments.

Both authors contributed equally to this work.

Citation

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Götz Kersting. Carmen Minuesa. "Defective Galton-Watson processes in a varying environment." Bernoulli 28 (2) 1408 - 1431, May 2022. https://doi.org/10.3150/21-BEJ1393

Information

Received: 1 November 2020; Revised: 1 July 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

Digital Object Identifier: 10.3150/21-BEJ1393

Keywords: absorption , branching process , defective distribution , family tree , varying environment

Rights: Copyright © 2022 ISI/BS

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Vol.28 • No. 2 • May 2022
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