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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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.

Article information

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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  • [1] R. Abraham and J.-F. Delmas. A construction of a $\beta$-coalescent via the pruning of binary trees. J. Appl. Probab. 50 (2013) 772–790.
  • [2] R. Abraham and J.-F. Delmas. $\beta$-Coalescents and stable Galton-Watson trees. ALEA Lat. Am. J. Probab. Math. Stat. 12 (2015) 451–476.
  • [3] R. Durrett. Probability: Theory and Examples, 4th edition. Cambridge University Press, Cambridge, 2010.
  • [4] B. Eldon and J. Wakeley. Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172 (2006) 2621–2633.
  • [5] A. Gnedin, A. Iksanov and A. Marynych. On $\Lambda$-coalescents with dust component. J. Appl. Probab. 48 (2011) 1133–1151.
  • [6] C. Goldschmidt and J. B. Martin. Random recursive trees and the Bolthausen–Sznitman coalescent. Electron. J. Probab. 10 (2005) 718–745.
  • [7] R. Grübel and K. Hagemann. Leader election: A Markov chain approach. Math. Appl. 44 (2016) 113–143.
  • [8] O. Hénard. The fixation line in the $\Lambda$-coalescent. Ann. Appl. Probab. 25 (2015) 3007–3032.
  • [9] A. E. Kyprianou. Fluctuations of Lévy Processes with Applications, 2nd edition. Springer, Heidelberg, 2014.
  • [10] M. Möhle. Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson–Dirichlet coalescent. Stochastic Process. Appl. 120 (2010) 2159–2173.
  • [11] M. Möhle. On hitting probabilities of beta coalescents and absorption times of coalescents that come down from infinity. ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 141–159.
  • [12] J. Pitman. Coalescents with multiple collisions. Ann. Probab. 27 (1999) 1870–1902.
  • [13] S. Sagitov. The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 (1999) 1116–1125.
  • [14] J. Schweinsberg. A necessary and sufficient condition for the $\Lambda$-coalescent to come down from infinity. Electron. Commun. Probab. 5 (2000) 1–11.