Sterile versus prolific individuals pertaining to linear-fractional Bienaymé–Galton–Watson trees

Abstract

In a Bienaymé–Galton–Watson process, for which there is a positive probability of individuals having no offspring, there is a subtle balance and dependence between the sterile nodes (the dead nodes or leaves) and the prolific nodes (the productive nodes), both at and up to the current generation. We explore the many facets of this problem, especially within the context of an exactly solvable linear-fractional branching mechanism, for which a distributional approach to asymptotics is made possible. The relation of this special branching process to skip-free to the left and simple random walks’ excursions is then investigated. Mutual statistical information on their shapes can be learnt from this association.