• Bernoulli
  • Volume 25, Number 1 (2019), 148-173.

Recovering the Brownian coalescent point process from the Kingman coalescent by conditional sampling

Amaury Lambert and Emmanuel Schertzer

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider a continuous population whose dynamics is described by the standard stationary Fleming–Viot process, so that the genealogy of $n$ uniformly sampled individuals is distributed as the Kingman $n$-coalescent. In this note, we study some genealogical properties of this population when the sample is conditioned to fall entirely into a subpopulation with most recent common ancestor (MRCA) shorter than $\varepsilon$. First, using the comb representation of the total genealogy (Lambert and Uribe Bravo (P-Adic Numbers Ultrametric Anal. Appl. 9 (2017) 22–38)), we show that the genealogy of the descendance of the MRCA of the sample on the timescale $\varepsilon$ converges as $\varepsilon\to0$. The limit is the so-called Brownian coalescent point process (CPP) stopped at an independent Gamma random variable with parameter $n$, which can be seen as the genealogy at a large time of the total population of a rescaled critical birth–death process, biased by the $n$th power of its size. Second, we show that in this limit the coalescence times of the $n$ sampled individuals are i.i.d. uniform random variables in $(0,1)$. These results provide a coupling between two standard models for the genealogy of a random exchangeable population: the Kingman coalescent and the Brownian CPP.

Article information

Bernoulli, Volume 25, Number 1 (2019), 148-173.

Received: November 2016
Revised: June 2017
First available in Project Euclid: 12 December 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

coalescent point process conditional sampling flows of bridges Kingman coalescent small time behavior


Lambert, Amaury; Schertzer, Emmanuel. Recovering the Brownian coalescent point process from the Kingman coalescent by conditional sampling. Bernoulli 25 (2019), no. 1, 148--173. doi:10.3150/17-BEJ971.

Export citation


  • [1] Berestycki, N. (2009). Recent Progress in Coalescent Theory. Ensaios Matemáticos [Mathematical Surveys] 16. Rio de Janeiro: Sociedade Brasileira de Matemática.
  • [2] Bertoin, J. (2006). Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics 102. Cambridge: Cambridge Univ. Press.
  • [3] Bertoin, J. and Le Gall, J.-F. (2003). Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 261–288.
  • [4] Bertoin, J. and Le Gall, J.-F. (2005). Stochastic flows associated to coalescent processes. II. Stochastic differential equations. Ann. Inst. Henri Poincaré Probab. Stat. 41 307–333.
  • [5] Bertoin, J. and Le Gall, J.-F. (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50 147–181.
  • [6] Cannings, C. (1974). The latent roots of certain Markov chains arising in genetics: A new approach. I. Haploid models. Adv. in Appl. Probab. 6 260–290.
  • [7] Donnelly, P. and Kurtz, T.G. (1999). Particle representations for measure-valued population models. Ann. Probab. 27 166–205.
  • [8] Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Series in Statistical and Probabilistic Mathematics 31. Cambridge: Cambridge Univ. Press.
  • [9] Ethier, S.N. and Kurtz, T.G. Markov Processes: Characterization and Convergence. Wiley Series in Probab. and Stat. Wiley.
  • [10] Fleming, W.H. and Viot, M. (1979). Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28 817–843.
  • [11] Kingman, J.F.C. (1982). The coalescent. Stochastic Process. Appl. 13 235–248.
  • [12] Knight, F.B. (1963). Random walks and a sojourn density process of Brownian motion. Trans. Amer. Math. Soc. 109 56–86.
  • [13] Lambert, A. (2008). Population dynamics and random genealogies. Stoch. Models 24 45–163.
  • [14] Lambert, A. (2010). The contour of splitting trees is a Lévy process. Ann. Probab. 38 348–395.
  • [15] Lambert, A. and Popovic, L. (2013). The coalescent point process of branching trees. Ann. Appl. Probab. 23 99–144.
  • [16] Lambert, A. and Uribe Bravo, G. (2017). The comb representation of compact ultrametric spaces. P-Adic Numbers Ultrametric Anal. Appl. 9 22–38.
  • [17] Le Gall, J.-F. (2005). Random trees and applications. Probab. Surv. 2 245–311.
  • [18] Möhle, M. and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29 1547–1562.
  • [19] Pfaffelhuber, P. and Wakolbinger, A. (2006). The process of most recent common ancestors in an evolving coalescent. Stochastic Process. Appl. 116 1836–1859.
  • [20] Pfaffelhuber, P., Wakolbinger, A. and Weisshaupt, H. (2011). The tree length of an evolving coalescent. Probab. Theory Related Fields 151 529–557.
  • [21] Popovic, L. (2004). Asymptotic genealogy of a critical branching process. Ann. Appl. Probab. 14 2120–2148.
  • [22] Rogers, L.C.G. and Williams, D. (1987). Diffusions, Markov Processes, and Martingales. Vol. 2: Itô Calculus. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley.