Abstract
We prove that a result of Haldane (1927) that relates the asymptotic behaviour of the extinction probability of a slightly supercritical Poisson branching process to the mean number of offspring is true for a general Bienaymé-Galton-Watson branching process, provided that the second derivatives of the probability-generating functions converge uniformly to a non-zero limit. We show also by examples that such a result is true more widely than our proof suggests and exhibit some counter-examples.
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Hoppe, F.M. Asymptotic rates of growth of the extinction probability of a mutant gene. J. Math. Biol. 30, 547–566 (1992). https://doi.org/10.1007/BF00948890
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DOI: https://doi.org/10.1007/BF00948890