Limit theorems for branching processes with mutations

Abstract : This thesis studies branching population models called splitting trees, where individuals evolve independently from one another, have independent and identically distributed lifetimes (that are not necessarily exponential), and give birth at constant rate during their lives. We further assume that each individual carries a type, and possibly undergoes a mutation at her birth, that changes her type into a new one. In the first chapter, we prove convegence results for bivariate Lévy processes with non negative jumps. These theoretical results are used in the second chapter to establish an invariance principle for the genealogical tree of the populations described above, enriched with their mutational history, in a large population size asymptotic. Finally we study in the third chapter the genealogical structure and the site frequency spectrum (number of mutations carried by a given number of individuals) for uniform samples in critical branching populations whose scaling limit is a Brownian tree (e.g., critical birth-death trees). Possible future applications of these results to population genetics are presented in the fourth chapter.
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Submitted on : Wednesday, October 15, 2014 - 10:15:52 AM
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Cécile Delaporte. Limit theorems for branching processes with mutations. General Mathematics [math.GM]. Université Pierre et Marie Curie - Paris VI, 2014. English. ⟨NNT : 2014PA066209⟩. ⟨tel-01074660⟩

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