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

Daniela Bertacchi, Peter Braunsteins, Sophie Hautphenne, Fabio Zucca
2022 Latin American Journal of Probability and Mathematical Statistics  
We consider Galton-Watson branching processes with countable typeset X . We study the vectors qpAq " pq x pAqq xPX recording the conditional probabilities of extinction in subsets of types A Ď X , given that the type of the initial individual is x. We first investigate the location of the vectors qpAq in the set of fixed points of the progeny generating vector and prove that q x ptxuq is larger than or equal to the xth entry of any fixed point, whenever it is different from 1. Next, we present
more » ... quivalent conditions for q x pAq ă q x pBq for any initial type x and A, B Ď X . Finally, we develop a general framework to characterise all distinct extinction probability vectors, and thereby to determine whether there are finitely many, countably many, or uncountably many distinct vectors. We illustrate our results with examples, and conclude with open questions.
doi:10.30757/alea.v19-12 fatcat:idaw3r6af5fvddsj3rjo652l4m