Abstract: Branching processes in varying environment (BPVEs) are a natural extension of Galton-Watson processes, where the distribution of the number of children depends on the generation. In this work, we define a notion of near criticality for BPVEs. We show that the genealogy of a near critial BPVE, viewed as a random metric space in the Gromov-Hausdorff-Prohorov topology, converges to a limiting tree. This limit is expressed as a time-change of the corresponding limit for Galton-Watson processes, the Brownian coalescent point process. This work also illustrates and extends a general approach to tackle the convergence of genealogies for branching processes. It relies on computing the so-called moments of the genealogy using a many-to-few formula.
The genealogy of a nearly critical branching process in varying environment
18.10.2022 16:45 - 17:45
Organiser:
M. Beiglböck, N. Berestycki, L. Erdös, J. Maas, F. Toninelli, E. Schertzer
Location: