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The results in this paper provide new information on asymptotic properties ofclassical models: the neutral Kingman coalescent under a generalfinite-alleles, parent-dependent mutation mechanism, and its generalisation,the ancestral selection graph. Several relevant quantities related to thesefundamental models are not explicitly known when mutations are parentdependent. Examples include the probability that a sample taken from apopulation has a certain type configuration, and the transition probabilitiesof their block counting jump chains. In this paper, asymptotic results arederived for these quantities, as the sample size goes to infinity. It is shownthat the sampling probabilities decay polynomially in the sample size withmultiplying constant depending on the stationary density of the Wright-Fisherdiffusion and that the transition probabilities converge to the limit offrequencies of types in the sample.

Asymptotic behaviour of sampling and transition probabilities in coalescent models under selection and parent dependent mutations