The size of the last merger and time reversal in $\Lambda$-coalescents

TL;DR

This paper analyzes the behavior of the last merger in $ ext{Lambda}$-coalescents, establishing conditions for different asymptotic regimes and exploring time-reversal properties of the process.

Contribution

It provides new conditions for the asymptotic behavior of the last merger size in $ ext{Lambda}$-coalescents, linking these to quasi-invariant measures and time-reversal dynamics.

Findings

Conditions for tightness, convergence to finite or infinite limits.

Relation of asymptotic behavior to quasi-invariant measures.

Analysis of time-reversal of the block-counting process.

Abstract

We consider the number of blocks involved in the last merger of a $\Lambda$-coalescent started with $n$ blocks. We give conditions under which, as $n \to \infty$, the sequence of these random variables a) is tight, b) converges in distribution to a finite random variable or c) converges to infinity in probability. Our conditions are optimal for $\Lambda$-coalescents that have a dust component. For general $\Lambda$, we relate the three cases to the existence, uniqueness and non-existence of quasi-invariant measures for the dynamics of the block-counting process, and in case b) investigate the time-reversal of the block-counting process back from the time of the last merger.