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The Ancestral Selection Graph (ASG) is an important genealogical processwhich extends the well-known Kingman coalescent to incorporate naturalselection. We show that the number of lineages of the ASG with and withoutmutation is asymptotic to $2/t$ as $t\to 0$, in agreement with the limitingbehaviour of the Kingman coalescent. We couple these processes on the sameprobability space using a Poisson random measure construction that allows us toprecisely compare their hitting times. These comparisons enable us tocharacterise the speed of coming down from infinity of the ASG as well as itsfluctuations in a functional central limit theorem. This extends similarresults for the Kingman coalescent.

Diffusion Limits at Small Times for Coalescent Processes with Mutation and Selection