In this work, we are interested in Growth of Galton-Watson trees under two different models: (1) Galton-Watson (GW) forests with lifetimes and/or immigrants, and (2) Galton-Watson forests with mutation, which we call Galton-Watson-Clone-Mutant forests, or GWCMforests. Under each model, we study certain consistent families (Fλ)λ≥0 of GW/GWCM forests and associated decompositions that include backbone decomposition as studied by many authors. Specifically, consistency here refers to the property that for each μ ≤ λ, the forest Fμ has the same distribution as the subforest of Fλ spanned by the blue leaves in a Bernoulli leaf colouring, where each leaf of Fλ is coloured in blue independently with probability μ/λ. In the first model, the case of exponentially distributed lifetimes and no immigration was studied by Duquesne and Winkel and related to the genealogy of Markovian continuous-state branching processes (CSBP). We characterise here such families in the framework of arbitrary lifetime distributions and immigration according to a renewal process, and show convergence to Sagitov’s (non-Markovian) generalisation of continuous-state branching renewal processes, and related processes with immigration. In the second model, we characterise such families in terms of certain bivariate CSBP with branching mechanisms studied previously by Watanabe and show associated convergence results. This is related to, but more general than Bertoin’s study of GWCM trees, and also ties in with work by Abraham and Delmas, who study directly some of the limiting processes.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:543558 |
| Date | January 2011 |
| Creators | Cao, Xiaoou |
| Contributors | Winkel, Matthias |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:beaa9fe1-d60c-4487-9520-e8f004b53e6f |
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