  The Ancestral Selection Graph (ASG) is an important genealogical process
which extends the well-known Kingman coalescent to incorporate natural
selection. We show that the number of lineages of the ASG with and without
mutation is asymptotic to $2/t$ as $t\to 0$, in agreement with the limiting
behaviour of the Kingman coalescent. We couple these processes on the same
probability space using a Poisson random measure construction that allows us to
precisely compare their hitting times. These comparisons enable us to
characterise the speed of coming down from infinity of the ASG as well as its
fluctuations in a functional central limit theorem. This extends similar
results for the Kingman coalescent.
