The common ancestor type distribution of a $\Lambda$-Wright-Fisher process with selection and mutation

Abstract

Using graphical methods based on a ‘lookdown’ and pruned version of the ancestral selection graph, we obtain a representation of the type distribution of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population. This extends results from [17] to the case of heavy-tailed offspring, directed by a reproduction measure $\Lambda$. The representation is in terms of the equilibrium tail probabilities of the line-counting process $L$ of the graph. We identify a strong pathwise Siegmund dual of $L$, and characterise the equilibrium tail probabilities of $L$ in terms of hitting probabilities of the dual process.