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

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 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.