Extinction probabilities in branching processes with countably many types: a general framework

D Bertacchi; P Braunsteins; S Hautphenne; F Zucca

Journal article

Extinction probabilities in branching processes with countably many types: a general framework

D Bertacchi, P Braunsteins, S Hautphenne, F Zucca

Alea Rio De Janeiro | Published : 2022

DOI: 10.30757/ALEA.V19-12

Abstract

We consider Galton–Watson branching processes with countable typeset X. We study the vectors (Formula Presented) recording the conditional probabilities of extinction in subsets of types (Formula Presented), given that the type of the initial individual is x. We first investigate the location of the vectors q(A) in the set of fixed points of the progeny generating vector and prove that (Formula Presented) is larger than or equal to the xth entry of any fixed point, whenever it is different from 1. Next, we present equivalent conditions for (Formula Presented) for any initial type x and (Formula Presented). Finally, we develop a general framework to characterise all distinct extinction probab..

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University of Melbourne Researchers

Sophie Hautphenne Author

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Funding Acknowledgements

The authors are grateful to the anonymous referee whose comments helped improve the manuscript. Daniela Bertacchi and Fabio Zucca acknowledge support from INDAM-GNAMPA and PRIN Grant 20155PAWZB. Peter Braunsteins has conducted part of the work while supported by the Australian Research Council (ARC) Laureate Fellowship FL130100039 and the Netherlands Organisation for Scientific Research (NWO) through Gravitationgrant NETWORKS-024.002.003. Sophie Hautphenne would like to thank the Australian Research Council (ARC) for support through her Discovery Early Career Researcher Award DE150101044 and her Discovery Project DP200101281. The authors also acknowledge the ARC Centre of Excellence for Mathematical and Statistical Frontiers (ACEMS) for supporting the research visit of Daniela Bertacchi and Fabio Zucca at The University of Melbourne, during which this work was initiated.