On the genealogy and coalescence times of Bienaymé–Galton–Watson branching processes. Issue 1 (2nd January 2018)

• Record Type: Journal Article
• Authors: Grosjean, Nicolas; Huillet, Thierry
• Is Part Of: Stochastic models. Volume 34:Issue 1(2018)
• Page Start: 1
• Page End: 24
• Publication Date: 2018-01-02
• Subjects: Ancestral process of the Bienyamé–Galton–Watson process -- coalescence times under conditioning -- sampling without replacement formulae; Primary 60J80; Stochastic processes -- Periodicals; Probabilities -- Periodicals; 519.2
• DOI: 10.1080/15326349.2017.1375958
• Languages: English
• ISSNs: 1532-6349

Abstract

ABSTRACT: Coalescence processes have received a lot of attention in the context of conditional branching processes with fixed population size and non-overlapping generations. Here, we focus on similar problems in the context of the standard unconditional Bienaymé–Galton–Watson branching processes, either (sub)-critical or supercritical. Using an analytical tool, we derive the structure of some counting aspects of the ancestral genealogy of such processes, including: the transition matrix of the ancestral count process and an integral representation of various coalescence times distributions, such as the time to most recent common ancestor of a random sample of arbitrary size, including full size. We illustrate our results on two important examples of branching mechanisms displaying either finite or infinite reproduction mean, their main interest being to offer a closed form expression for their probability generating functions at all times. Large time behaviors are investigated.