Electronic Journal of Probability

Diffusion limits at small times for $\Lambda$-coalescents with a Kingman component

Vlada Limic and Anna Talarczyk

Full-text: Open access


We consider  standard $\Lambda$-coalescents (or coalescents with multiple collisions) with a non-trivial "Kingman part". That is,  the driving measure $\Lambda$ has an atom at $0; \Lambda(\{0\})= c > 0$. It is known that all such coalescents come down from infinity. Moreover, the  number of blocks $N_t$  is asymptotic to $v(t) = 2/(ct)$ as $t\to 0$. In the present paper we investigate  the second-order asymptotics of $N_t$ in the functional sense at small times. This  complements our earlier results on the fluctuations of the number of blocks for a class of regular $\Lambda$-coalescents without the Kingman part. In the present setting it turns out that  the Kingman part dominates and the limit process is a Gaussian diffusion, as opposed to the stable limit in our previous work.

Article information

Electron. J. Probab. Volume 20 (2015), paper no. 45, 20 pp.

Accepted: 18 April 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60F17: Functional limit theorems; invariance principles 92D25: Population dynamics (general) 60J60: Diffusion processes [See also 58J65] 60G55: Point processes

Kingman coalescent $\Lambda$-coalescent coming down from infinity functional limit theorems diffusion processes Poisson random measure

This work is licensed under a Creative Commons Attribution 3.0 License.


Limic, Vlada; Talarczyk, Anna. Diffusion limits at small times for $\Lambda$-coalescents with a Kingman component. Electron. J. Probab. 20 (2015), paper no. 45, 20 pp. doi:10.1214/EJP.v20-3818. https://projecteuclid.org/euclid.ejp/1465067151

Export citation


  • Aldous, David J. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5 (1999), no. 1, 3–48.
  • Berestycki, Julien; Berestycki, Nathanaël; Limic, Vlada. The $\Lambda$-coalescent speed of coming down from infinity. Ann. Probab. 38 (2010), no. 1, 207–233.
  • Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason. Small-time behavior of beta coalescents. Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 2, 214–238.
  • Berestycki, Nathanaël. Recent progress in coalescent theory. Ensaios Matemáticos [Mathematical Surveys], 16. Sociedade Brasileira de Matemática, Rio de Janeiro, 2009. 193 pp. ISBN: 978-85-85818-40-1
  • Bertoin, Jean. Random fragmentation and coagulation processes. Cambridge Studies in Advanced Mathematics, 102. Cambridge University Press, Cambridge, 2006. viii+280 pp. ISBN: 978-0-521-86728-3; 0-521-86728-2
  • Bertoin, Jean; Le Gall, Jean-Francois. Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 (2006), no. 1-4, 147–181 (electronic).
  • Bojdecki, Tomasz; Gorostiza, Luis G.; Talarczyk, Anna. Number variance for hierarchical random walks and related fluctuations. Electron. J. Probab. 16 (2011), no. 75, 2059–2079.
  • Dahmer, Iulia; Kersting, Götz; Wakolbinger, Anton. The total external branch length of Beta-coalescents. Combin. Probab. Comput. 23 (2014), no. 6, 1010–1027.
  • Donnelly, Peter; Kurtz, Thomas G. Particle representations for measure-valued population models. Ann. Probab. 27 (1999), no. 1, 166–205.
  • Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8
  • Griffiths, R. C. Asymptotic line-of-descent distributions. J. Math. Biol. 21 (1984), no. 1, 67–75.
  • Kersting, Götz. The asymptotic distribution of the length of beta-coalescent trees. Ann. Appl. Probab. 22 (2012), no. 5, 2086–2107.
  • Kersting, Götz; Schweinsberg, Jason; Wakolbinger, Anton. The evolving beta coalescent. Electron. J. Probab. 19 (2014), no. 64, 27 pp.
  • Kingman, J. F. C. The coalescent. Stochastic Process. Appl. 13 (1982), no. 3, 235–248.
  • Kingman, J. F. C. On the genealogy of large populations. Essays in statistical science. J. Appl. Probab. 1982, Special Vol. 19A, 27–43.
  • V. Limic and A. Talarczyk. Second-order asymptotics for the block counting process in a class of regularly varying Λ-coalescents, Ann. Probab. (to appear), ARXIV1304.5183
  • Peszat, S.; Zabczyk, J. Stochastic partial differential equations with Lévy noise. An evolution equation approach. Encyclopedia of Mathematics and its Applications, 113. Cambridge University Press, Cambridge, 2007. xii+419 pp. ISBN: 978-0-521-87989-7
  • Pitman, Jim. Coalescents with multiple collisions. Ann. Probab. 27 (1999), no. 4, 1870–1902.
  • Sagitov, Serik. The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 (1999), no. 4, 1116–1125.
  • Schweinsberg, Jason. A necessary and sufficient condition for the $\Lambda$-coalescent to come down from infinity. Electron. Comm. Probab. 5 (2000), 1–11 (electronic).
  • Schweinsberg, Jason. Dynamics of the evolving Bolthausen-Sznitman coalecent. Electron. J. Probab. 17 (2012), no. 91, 50 pp.