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

Abstract

The Kingman coalescent is an important and well studied process in population genetics modelling the ancestry of a sample of individuals backwards in time. In this paper, weak convergence is proved for a sequence of Markov chains consisting of two components related to the Kingman coalescent, as the size of the initial configuration, the sample size, grows to infinity. The first component is the normalised jump chain of the block counting processes of the Kingman coalescent with a finite number of d genetic types. The second component is a d 2 -dimensional process counting the number of mutations between types occurring in the Kingman coalescent. Time is scaled by the sample size. The limiting process consists of a deterministic d-dimensional component describing the limit of the jump chain and d 2 independent Poisson processes with state-dependent intensities, exploding at the origin, describing the limit of the number of mutations. The weak convergence result is first proved, using a generator approach, in the setting of parent independent mutations. A change of measure argument is used to extend the weak convergence result to include parent dependent mutations.

Asymptotic behaviour of sampling and transition probabilities in coalescent models under selection and parent dependent mutations

“…Thus a theoretical large-sample-size efficiency analysis of algorithms based on coalescent models, such as the ones in [22,23,7,2,12,13,3,16,17], would be useful. The study of large-sample-size asymptotic properties of the coalescent, to which this paper and [11] aim to contribute, are relevant for such analysis. Furthermore, large-sample-size asymptotic results provide tools for the analysis of differences between the coalescent approximation and the original model.…”