Asymptotics of the frequency spectrum for general Dirichlet Xi-coalescents

TL;DR

This paper investigates the asymptotic behavior of the frequency spectrum in general Dirichlet Xi-coalescents, revealing how these models behave under rescaling and their implications for population genetics statistics.

Contribution

It introduces a new analysis of Dirichlet Xi-coalescents, showing their convergence to a Markov Additive Process and deriving asymptotics for the frequency spectrum.

Findings

Rescaled block counting process converges to a martingale problem solution.

The limiting process's Lamperti transform is a Markov Additive Process.

Rescaled number of mutations converges to an exponential functional of a subordinator.

Abstract

In this work, we study general Dirichlet coalescents, which are a family of Xi-coalecents constructed from i.i.d mass partitions, and are an extension of the symmetric coalescent. This class of models is motivated by population models with recurrent demographic bottlenecks. We study the short time behavior of the multidimensional block counting process whose i-th component counts the number of blocks of size i. Compared to standard coalescent models (such as the class of Lambda-coalescents coming down from infinity), our process has no deterministic speed of coming down from infinity. In particular, we prove that, under appropriate re-scaling, it converges to a stochastic process which is the unique solution of a martingale problem. We show that the multivariate Lamperti transform of this limiting process is a Markov Additive Process (MAP). This allows us to provide some asymptotics for the n-Site Frequency Spectrum, which is a statistic widely used in population genetics. In particular, the rescaled number of mutations converges to the exponential functional of a subordinator.