Weak Convergence of Non-neutral Genealogies to Kingman's Coalescent

TL;DR

This paper proves that genealogies in non-neutral genetic models converge weakly to Kingman's coalescent, allowing comprehensive analysis of genealogical trees in evolutionary and Monte Carlo systems.

Contribution

It extends convergence results from finite-dimensional distributions to the entire genealogical process under non-neutral conditions.

Findings

Weak convergence of genealogies established

Enables analysis of full genealogical trees

Applicable to genetic evolution and Monte Carlo methods

Abstract

Interacting particle systems undergoing repeated mutation and selection steps model genetic evolution, and also describe a broad class of sequential Monte Carlo methods. The genealogical tree embedded into the system is important in both applications. Under neutrality, when fitnesses of particles are independent from those of their parents, rescaled genealogies are known to converge to Kingman's coalescent. Recent work has established convergence under non-neutrality, but only for finite-dimensional distributions. We prove weak convergence of non-neutral genealogies on the space of c\`adl\`ag paths under standard assumptions, enabling analysis of the whole genealogical tree.