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

TL;DR

This paper analyzes the small-time behavior of the Ancestral Selection Graph (ASG), extending known results from the Kingman coalescent to include mutation and selection, and characterizes its convergence and fluctuations.

Contribution

It provides the first detailed asymptotic analysis of the ASG with mutation and selection at small times, including coupling constructions and fluctuation results.

Findings

Number of lineages asymptotic to 2/t as t→0

Coupling of ASG with and without mutation using Poisson measures

Characterization of the coming down from infinity and fluctuations

Abstract

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.