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The Contour Process of General Branching Processes

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.

Earlier Event: March 25
Lunch Break