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


Bertoin, J (2010). A limit theorem for trees of alleles in branching processes with rare neutral mutations. Stochastic Processes and their Applications, 120(5):678-697.

Abstract

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 Jiřina process) in discrete time. Itô’s excursion theory and the Lévy–Itô decomposition of subordinators provide fundamental insights for the results.

Abstract

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 Jiřina process) in discrete time. Itô’s excursion theory and the Lévy–Itô decomposition of subordinators provide fundamental insights for the results.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords: Weak convergence; Branching process; Neutral mutations; Allelic partition; Lévy–Itô decomposition
Language:English
Date:May 2010
Deposited On:25 Apr 2013 10:02
Last Modified:05 Apr 2016 16:44
Publisher:Elsevier
ISSN:0304-4149
Free access at:Related URL. An embargo period may apply.
Publisher DOI:https://doi.org/10.1016/j.spa.2010.01.017
Related URLs:http://arxiv.org/abs/0904.0581

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