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

摘要

We are interested in the genealogical structure of alleles for aBienaym\'e-Galton-Watson branching process with neutral mutations (infinitealleles model), in the situation where the initial population is large and themutation rate small. We shall establish that for an appropriate regime, theprocess of the sizes of the allelic sub-families converges in distribution to acertain continuous state branching process (i.e. a Jirina process) in discretetime. It\^o's excursion theory and the L\'eevy-It\^o decomposition ofsubordinators provide fundamental insights for the results.