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August 2018 The size of the last merger and time reversal in $\Lambda$-coalescents
Götz Kersting, Jason Schweinsberg, Anton Wakolbinger
Ann. Inst. H. Poincaré Probab. Statist. 54(3): 1527-1555 (August 2018). DOI: 10.1214/17-AIHP847

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 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.

Nous considérons le nombre de blocs impliqués dans le dernier regroupement d’un $\Lambda$-coalescent issu de $n$ blocs. Nous donnons des conditions sous lesquelles, quand $n$ tend vers l’infini, la suite de variables aléatoires (a) est tendue (b) converge en loi vers une variable aléatoire finie ou (c) converge vers l’infini en probabilité. Nos conditions sont optimales pour les $\Lambda$-coalescents qui ont une composante de poussière. Pour un $\Lambda$ général, nous associons ces trois cas à l’existence, l’unicité et la non-existence d’une mesure invariante pour la dynamique du processus de comptage des blocs. Dans le cas (b), nous étudions le retourné en temps du processus de comptage des blocs depuis de le temps de dernier regroupement.

Citation

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Götz Kersting. Jason Schweinsberg. Anton Wakolbinger. "The size of the last merger and time reversal in $\Lambda$-coalescents." Ann. Inst. H. Poincaré Probab. Statist. 54 (3) 1527 - 1555, August 2018. https://doi.org/10.1214/17-AIHP847

Information

Received: 3 January 2017; Revised: 16 May 2017; Accepted: 23 May 2017; Published: August 2018
First available in Project Euclid: 11 July 2018

zbMATH: 06976084
MathSciNet: MR3825890
Digital Object Identifier: 10.1214/17-AIHP847

Subjects:
Primary: 60J27
Secondary: 60G51, 60K05

Rights: Copyright © 2018 Institut Henri Poincaré

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Vol.54 • No. 3 • August 2018
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