Partly exchangeable fragmentations

Thesis

We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this thesis, we obtain their splitting rules, dislocation measures both in ranked order and in sized-biased order, and we study their limiting behaviour. We further extend the underlying exchangeable fragmentation processes of such trees into partly exchangeable fragmentation processes by weakening the exchangeability. We obtain the integral representations for the measures associated with partly exchangeable fragmentation processes and subordinator of the tagged fragments. We also embed the trees associated with such processes into continuum random trees and study their limiting behaviour. In the end, we generate a three-parameter family of partly exchangeable trees which contains the family of the alpha-gamma trees and another important two-parameter family based on Poisson-Dirichlet distributions.

Authors: Chen, B

• Publication date: 2009
• DOI: 10.5287/ora-e2z9oobo9
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Poisson-Dirichlet distribution; splitting rule; Markov branching model; self-similar fragmentation; R-tree; dislocation measure; continuum random tree; part exchangeability; alpha-gamma tree; exchangeability; sampling consistency
• Subjects: Mathematical genetics and bioinformatics (statistics); Probability theory and stochastic processes; Probability; Stochastic processes