Weak convergence of the scaled jump chain and number of mutations of the Kingman coalescent

TL;DR

This paper investigates the asymptotic behavior of the Kingman coalescent with mutations in large samples, showing weak convergence of scaled components to deterministic and Poisson processes, and introduces a new method for handling complex mutation schemes.

Contribution

It provides the first weak convergence results for the coalescent with general finite-alleles mutations, including parent dependent mutations, using a novel change of measure approach.

Findings

Scaled block-counting components converge to deterministic limits.

Mutation-counting components converge to Poisson processes.

The new approach extends results to parent dependent mutation models.

Abstract

The Kingman coalescent is a fundamental process in population genetics modelling the ancestry of a sample of individuals backwards in time. In this paper, in a large-sample-size regime, we study asymptotic properties of the coalescent under neutrality and a general finite-alleles mutation scheme, i.e. including both parent independent and parent dependent mutation. In particular, we consider a sequence of Markov chains that is related to the coalescent and consists of block-counting and mutation-counting components. We show that these components, suitably scaled, converge weakly to deterministic components and Poisson processes with varying intensities, respectively. Along the way, we develop a novel approach, based on a change of measure, to generalise the convergence result from the parent independent to the parent dependent mutation setting, in which several crucial quantities are not known explicitly.