Diffusion limits at small times for coalescents with a Kingman component

TL;DR

This paper analyzes the small-time behavior of the number of blocks in $ ext{Lambda}$-coalescents with a Kingman component, revealing Gaussian diffusion limits that extend understanding of their second-order asymptotics.

Contribution

It provides the first detailed study of second-order asymptotics for coalescents with a Kingman part, showing Gaussian limits dominate at small times.

Findings

Number of blocks $N_t$ asymptotic to $2/(ct)$ as $t o 0$

Limit process is a Gaussian diffusion, not stable

Kingman component dominates small-time fluctuations

Abstract

We consider standard $\La$-coalescents (or coalescents with multiple collisions) with a non-trivial "Kingman part". Equivalently, the driving measure $\Lambda$ has an atom at $0$; $\Lambda(\{0\})=c>0$. It is known that all such coalescents come down from infinity. Moreover, the number of blocks $N_t$ is asymptotic to $v(t) = 2/(ct)$ as $t\to 0$. In the present paper we investigate the second-order asymptotics of $N_t$ in the functional sense at small times. This complements our earlier results on the fluctuations of the number of blocks for a class of regular $\La$-coalescents without the Kingman part. In the present setting it turns out that the Kingman part dominates, and the limit process is a Gaussian diffusion, as opposed to the stable limit in our previous work.