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.
Subject(s)
Gene Frequency/genetics , Mutation/genetics , Probability , Genes/genetics , Mathematics , Models, GeneticABSTRACT
We prove a general result about the asymptomatic behaviour of the survival probability of a slightly supercritical multitype Bienaymé-Galton-Watson branching process. This is the complete analogue of a result which Ewens (1968) obtained for a Poisson branching process.
Subject(s)
Genetics, Population , Mutation/genetics , Models, Genetic , ProbabilityABSTRACT
The behaviour of a Pólya-like urn which generates Ewens' sampling formula in population genetics is investigated. Connections are made with work of Watterson and Kingman and to the Poisson-Dirichlet distribution. The order in which novel types occur in the urn is shown to parallel the age distribution of the infinitely many alleles diffusion model and consequences of this property are explored. Finally the urn process is related to Kingman's coalescent with mutation to provide a rigorous basis for this parallel.
Subject(s)
Alleles , Genetics, Population , Models, Genetic , Biometry , Random AllocationABSTRACT
Any direct estimator of mutation in a human population is subject to error due to nonpaternity. This paper deals with the quantification of this error by producing, under certain assumptions, the probability for paternity. In addition, a new direct estimator of the mutation rate is introduced.
