Advances in Applied Probability

Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration

Clément Foucart

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


Coalescents with multiple collisions (also called Λ-coalescents or simple exchangeable coalescents) are used as models of genealogies. We study a new class of Markovian coalescent processes connected to a population model with immigration. Consider an infinite population with immigration labelled at each generation by N := {1, 2, ...}. Some ancestral lineages cannot be followed backwards after some time because their ancestor is outside the population. The individuals with an immigrant ancestor constitute a distinguished family and we define exchangeable distinguished coalescent processes as a model for genealogy with immigration, focusing on simple distinguished coalescents, i.e. such that when a coagulation occurs all the blocks involved merge as a single block. These processes are characterized by two finite measures on [0, 1] denoted by M = (Λ0, Λ1). We call them M-coalescents. We show by martingale arguments that the condition of coming down from infinity for the M-coalescent coincides with that obtained by Schweinsberg for the Λ-coalescent. In the same vein as Bertoin and Le Gall, M-coalescents are associated with some stochastic flows. The superprocess embedded can be viewed as a generalized Fleming-Viot process with immigration. The measures Λ0 and Λ1 respectively specify the reproduction and the immigration. The coming down from infinity of the M-coalescent will be interpreted as the initial types extinction: after a certain time all individuals are immigrant children.

Article information

Adv. in Appl. Probab. Volume 43, Number 2 (2011), 348-374.

First available in Project Euclid: 21 June 2011

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces 60G09: Exchangeability
Secondary: 92D25: Population dynamics (general)

Exchangeable partition coalescent theory genealogy for a population with immigration stochastic flow coming down from infinity


Foucart, Clément. Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration. Adv. in Appl. Probab. 43 (2011), no. 2, 348--374.

Export citation


  • Aldous, D. J. (1985). Exchangeability and related topics. In École d'été de Probabilités de Saint-Flour, XIII-1983 (Lecture Notes Math. 1117), Springer, Berlin, pp. 1–198.
  • Berestycki, J., Berestycki, N. and Limic, V. (2011). Asymptotic sampling formulae and particle system representations for $\Lambda$-coalescents. Submitted.
  • Berestycki, N. (2010). Recent progress in coalescent theory. Math. Surveys 16, 193pp.
  • Bertoin, J. (2006). Random Fragmentation and Coagulation Processes (Camb. Stud. Adv. Math. 102), Cambridge University Press.
  • Bertoin, J. and Le Gall, J.-F. (2003). Stochastic flows associated to coalescent processes. Prob. Theory Relat. Fields 126, 261–288.
  • Bertoin, J. and Le Gall, J.-F. (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50, 147–181.
  • Birkner, M. et al. (2005). Alpha-stable branching and beta-coalescents. Electron. J. Prob. 10, 303–325.
  • Donnelly, P. and Joyce, P. (1991). Consistent ordered sampling distributions: characterization and convergence. Adv. Appl. Prob. 23, 229–258.
  • Donnelly, P. and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Prob. 27, 166–205.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.
  • Gnedin, A. V. (1997). The representation of composition structures. Ann. Prob. 25, 1437–1450.
  • Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Theory Prob. Appl. 16, 36–54.
  • Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Prob. Theory Relat. Fields 102, 145–158.
  • Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870–1902.
  • Pitman, J. (2006). Combinatorial Stochastic Processes (Lecture Notes Math. 1875), Springer, Berlin.
  • Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116–1125.
  • Schweinsberg, J. (2000). A necessary and sufficient condition for the $\Lambda$-coalescent to come down from infinity. Electron. Commun. Prob. 5, 1–11.
  • Schweinsberg, J. (2000). Coalescents with simultaneous multiple collisions. Electron. J. Prob. 5, 50pp.