By Emmanuel Schertzer, Université Paris VI
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 talk, I will focus on the height and contour processes encoding a general CMJ forest.
I will first show that the height process can be expressed in terms of a random transformation of the ladder height process associated with the underlying Lukasiewicz path. I will present two applications of this result: (1) in the case of ``short'' edges, the height process of a CMJ is obtained by stretching by a constant factor the height process of the associated genealogical Galton–Watson tree, and (2) when the offspring distribution has a finite second moment, the genealogy of the CMJ can be obtained from the underlying genealogical structure by a marking procedure, related to the so-called Poisson snake.
This is joint work with Florian Simatos.