Genealogical constructions and asymptotics for continuous-time Markov and continuous-state branching processes. (1st February 2019)

• Record Type: Journal Article
• Authors: Kurtz, Thomas G.
• Editors: Asmussen, S.; Klebaner, F.; Nerman, O.; Vatutin, V.
• Is Part Of: Advances in applied probability. Volume 50:Number A(2018)
• Page Start: 197
• Page End: 209
• Publication Date: 2019-02-01
• Subjects: Genealogical construction, -- Markov branching process, -- supercritical limit, -- Seneta‒Heyde norming, -- continuous-state branching process; Primary 60J80, -- Secondary 60F15, -- 60J25; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2018.80
• Languages: English
• ISSNs: 0001-8678

Abstract

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.