Height and contour processes of Crump-Mode-Jagers forests (II): The Bellman-Harris universality class

Emmanuel Schertzer (1, 2) , Florian Simatos (3)

1 UPMC - Université Pierre et Marie Curie - Paris 6
2 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
3 DISC - Département d'Ingénierie des Systèmes Complexes

Crump-Mode-Jagers (CMJ) trees generalize Galton-Watson trees by allowing individuals to live for an arbitrary duration and give birth at arbitrary times during their life-time. In this paper, we exhibit a simple condition under which the height and contour processes of CMJ forests belong to the universality class of Bellman-Harris processes. This condition formalizes an asymptotic independence between the chronological and genealogical structures. We show that it is satisfied by a large class of CMJ processes and in particular, quite surprisingly, by CMJ processes with a finite variance offspring distribution. Along the way, we prove a general tightness result.

Mots clés: Crump-Mode-Jagers branching processes; Chronologial trees; Scaling limits; Invariance principles

Domaines: Autre

Emmanuel Schertzer, Florian Simatos. Height and contour processes of Crump-Mode-Jagers forests (II): The Bellman-Harris universality class. Electronic Journal of Probability, 2019, 24, pp.47. ⟨10.1214/19-EJP307⟩. ⟨hal-02168469⟩