A limit theorem for trees of alleles in branching processes with rare neutral mutations

Jean Bertoin (1, 2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We are interested in the genealogical structure of alleles for a Bienaymé-Galton-Watson branching process with neutral mutations (infinite alleles model), in the situation where the initial population is large and the mutation rate small. We shall establish that for an appropriate regime, the process of the sizes of the allelic sub-families converges in distribution to a certain continuous state branching process (i.e. a Jirina process) in discrete time. Itô's excursion theory and the Léevy-Itô decomposition of subordinators provide fundamental insights for the results.

Mots clés: Lévy-Itô decomposition

Mots clés: Weak convergence; branching process; neutral mutations; allelic partition; Lévy-Itô decomposition.

Domaines: Probabilités [math.PR]

Jean Bertoin. A limit theorem for trees of alleles in branching processes with rare neutral mutations. 2009. ⟨hal-00373262v2⟩