Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-09T07:41:31.605Z Has data issue: false hasContentIssue false
  • English
  • Français

Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Clément Foucart
Clément Foucart*
Affiliation:
Université Pierre et Marie Curie
*
∗ Postal address: Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Exchangeable partitioncoalescent theorygenealogy for a population with immigrationstochastic flowcoming down from infinity

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces 60G09: Exchangeability
Secondary: 92D25: Population dynamics (general)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 2 , June 2011 , pp. 348 - 374
Copyright
Copyright © Applied Probability Trust 2011 

References

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.Google Scholar
Berestycki, J., Berestycki, N. and Limic, V. (2011). Asymptotic sampling formulae and particle system representations for Λ-coalescents. Submitted.Google Scholar
Berestycki, N. (2010). Recent progress in coalescent theory. Math. Surveys 16, 193pp.Google Scholar
Bertoin, J. (2006). Random Fragmentation and Coagulation Processes (Camb. Stud. Adv. Math. 102), Cambridge University Press.Google Scholar
Bertoin, J. and Le Gall, J.-F. (2003). Stochastic flows associated to coalescent processes. Prob. Theory Relat. Fields 126, 261–288.Google Scholar
Bertoin, J. and Le Gall, J.-F. (2006). Stochastic flows associated to coalescent processes. III. Limit theorems. Illinois J. Math. 50, 147–181.Google Scholar
Birkner, M. et al. (2005). Alpha-stable branching and beta-coalescents. Electron. J. Prob. 10, 303–325.Google Scholar
Donnelly, P. and Joyce, P. (1991). Consistent ordered sampling distributions: characterization and convergence. Adv. Appl. Prob. 23, 229–258.Google Scholar
Donnelly, P. and Kurtz, T. G. (1999). Particle representations for measure-valued population models. Ann. Prob. 27, 166–205.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. John Wiley, New York.Google Scholar
Gnedin, A. V. (1997). The representation of composition structures. Ann. Prob. 25, 1437–1450.Google Scholar
Kawazu, K. and Watanabe, S. (1971). Branching processes with immigration and related limit theorems. Theory Prob. Appl. 16, 36–54.Google Scholar
Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Prob. Theory Relat. Fields 102, 145–158.Google Scholar
Pitman, J. (1999). Coalescents with multiple collisions. Ann. Prob. 27, 1870–1902.Google Scholar
Pitman, J. (2006). Combinatorial Stochastic Processes (Lecture Notes Math. 1875), Springer, Berlin.Google Scholar
Sagitov, S. (1999). The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Prob. 36, 1116–1125.Google Scholar
Schweinsberg, J. (2000). A necessary and sufficient condition for the Λ-coalescent to come down from infinity. Electron. Commun. Prob. 5, 1–11.Google Scholar
Schweinsberg, J. (2000). Coalescents with simultaneous multiple collisions. Electron. J. Prob. 5, 50pp.Google Scholar