Sterile versus prolific individuals pertaining to linear-fractional Bienaym{\'e}-Galton-Watson trees

TL;DR

This paper investigates the relationship between sterile and prolific individuals in linear-fractional Bienaymé-Galton-Watson trees, revealing their dependence and connections to random walks, with implications for understanding branching processes.

Contribution

It provides an exact analysis of the dependence between sterile and prolific nodes in linear-fractional branching processes and explores their relation to random walk excursions.

Findings

Dependence between sterile and prolific nodes characterized

Asymptotic behaviors of the process analyzed

Connections to random walk excursions established

Abstract

In a Bienaym\'{e}-Galton-Watson process for which there is a positiveprobability for individuals of having no offspring, there is a subtlebalance and dependence between the sterile nodes (the dead nodes or leaves)and the prolific ones (the productive nodes) both at and up to the currentgeneration. We explore the many facets of this problem, especially in thecontext of an exactly solvable linear-fractional branching mechanism at allgeneration. Eased asymptotic issues are investigated. Relation of thisspecial branching process to skip-free to the left and simple random walks'excursions is then investigated. Mutual statistical information on theirshapes can be learnt from this association.