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Bertoin, Jean

A limit theorem for trees of alleles in branching processes with rare neutral mutations. (English) Zbl 1193.60099

Stochastic Processes Appl. 120, No. 5, 678-697 (2010).
Summary: 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.

Cited in 2 Reviews
Cited in 9 Documents

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J05 Discrete-time Markov processes on general state spaces

Keywords:

weak convergence; branching process; neutral mutations; allelic partition; Lévy-Itô decomposition
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\textit{J. Bertoin}, Stochastic Processes Appl. 120, No. 5, 678--697 (2010; Zbl 1193.60099)
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References:

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