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

Götz Kersting; Jason Schweinsberg; Anton Wakolbinger

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arXiv · 1701.00549

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

Götz Kersting·Jason Schweinsberg·Anton Wakolbinger

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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.

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Götz Kersting, Jason Schweinsberg, Anton Wakolbinger. 2017-01-02. The size of the last merger and time reversal in $\Lambda$-coalescents. https://arxiv.org/abs/1701.00549

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