Genealogical constructions and asymptotics for continuous-time Markov and continuous-state branching processes. (1st February 2019) - Kurtz, Thomas G.
Abstract: Genealogical constructions of population processes provide models which simultaneously record the forward-in-time evolution of the population size (and distribution of locations and types for models that include them) and the backward-in-time genealogies of the individuals in the population at each time t . A genealogical construction for continuous-time Markov branching processes from Kurtz and Rodrigues (2011) is described and exploited to give the normalized limit in the supercritical case. A Seneta‒Heyde norming is identified as a solution of an ordinary differential equation. The analogous results are given for continuous-state branching processes, including proofs of the normalized limits of Grey (1974) in both the supercritical and critical/subcritical cases.