Advances in Applied Probability

Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration

Clément Foucart

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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

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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.

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  • 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.