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

• Published in: Electronic journal of probability Vol. 29; no. none
• Main Authors: Favero, Martina, Hult, Henrik
• Format: Journal Article
• Language: English
• Published: 01.01.2024
• Subjects: coalescent; large sample size; parent dependent mutations; population genetics; weak convergence
• ISSN: 1083-6489; 1083-6489
• DOI: 10.1214/24-EJP1128

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 largesample -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 mutationcounting 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.