Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Götz Kersting, Jason Schweinsberg, and 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 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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 3 (2018), 1527-1555.

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1531296028

Digital Object Identifier
doi:10.1214/17-AIHP847

Mathematical Reviews number (MathSciNet)
MR3825890

Zentralblatt MATH identifier
06976084

Subjects
Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60K05: Renewal theory 60G51: Processes with independent increments; Lévy processes

Keywords
$\Lambda$-coalescent Block-counting process Renewal theory Subordinator

Citation

Kersting, Götz; Schweinsberg, Jason; Wakolbinger, Anton. The size of the last merger and time reversal in $\Lambda$-coalescents. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 3, 1527--1555. doi:10.1214/17-AIHP847. https://projecteuclid.org/euclid.aihp/1531296028


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