Ornstein-Uhlenbeck fluctuations for the line counting process of the ancestral selection graph

TL;DR

This paper proves that, under strong or moderate selection in the Moran model, the fluctuations of the ancestral selection graph's line counting process converge to an Ornstein-Uhlenbeck process, extending existing functional limit theorems.

Contribution

It extends Ethier and Kurtz's 1986 functional limit theorem to analyze fluctuations in the ancestral selection graph under selection.

Findings

Fluctuations converge to an Ornstein-Uhlenbeck process.

Extension of a key functional limit theorem.

Applicable to logistic branching processes.

Abstract

For the Moran model with strong or moderately strong selection we prove that the fluctuations around the deterministic limit of the line counting process of the ancestral selection graph converge to an Ornstein-Uhlenbeck process. To this purpose we provide an extension of a functional limit theorem by Ethier and Kurtz 1986. This result and a small adaptation of our arguments can also be used to obtain the scaling limit for the fluctuations of certain logistic branching processes.