Asymptotic number of caterpillars of regularly varying $\Lambda$-coalescents that come down from infinity

TL;DR

This paper investigates the asymptotic behavior of the number of r-caterpillars in $ ext{Lambda}$-coalescents that come down from infinity, establishing convergence results under regular variation assumptions.

Contribution

It provides the first explicit asymptotic characterization of the number of r-caterpillars in $ ext{Lambda}$-coalescents with regular variation, including convergence to a constant.

Findings

Number of r-caterpillars converges to an explicit constant as sample size increases

Results apply to $ ext{Lambda}$-coalescents that come down from infinity

Provides asymptotic formulas under regular variation assumptions

Abstract

In this paper we look at the asymptotic number of r-caterpillars for $\Lambda$-coalescents which come down from infinity, under a regularly varying assumption. An r-caterpillar is a functional of the coalescent process started from $n$ individuals which, roughly speaking, is a block of the coalescent at some time, formed by one line of descend to which r-1 singletons have merged one by one. We show that the number of $r$-caterpillars, suitably scaled, converge to an explicit constant as the sample size n goes to infinity.