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4 - Exchangeable coalescents

Published online by Cambridge University Press:  07 December 2009

Jean Bertoin
Affiliation:
Université de Paris VI (Pierre et Marie Curie)
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Summary

Exchangeable coalescents form a natural family of Markov processes with values in the space of partitions of ℕ, in which blocks coagulate as time passes. We shall first present the celebrated coalescent of Kingman, which is often used as a model for the genealogy of large populations. Then we shall introduce a more general family of coalescents, called here exchangeable coalescents, in which coagulations may involve several blocks simultaneously. We shall also investigate the coagulation processes of mass-partitions associated with such exchangeable coalescents. The last section of this chapter is devoted to a representation of these exchangeable coalescents in terms of certain stochastic flows on the unit interval. Many ideas and techniques based on exchangeability, which were useful for investigating fragmentations, can also be fruitfully applied to coagulations.

Kingman's coalescent

Coalescence naturally arises when one studies the genealogy of populations; we first briefly explain why. Following Kingman [139], this will lead us to introduce a natural Markov process with values in the space P of partitions of ℕ.

Genealogy of populations in the Wright-Fisher model

Imagine at time T > 0 a population with size n which can be identified with the set [n] = {1, …, n}. Assume the population is haploid, meaning that each individual has exactly one parent at the previous generation, so we may follow its ancestral lineage backwards in time.

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Publisher: Cambridge University Press
Print publication year: 2006

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  • Exchangeable coalescents
  • Jean Bertoin, Université de Paris VI (Pierre et Marie Curie)
  • Book: Random Fragmentation and Coagulation Processes
  • Online publication: 07 December 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511617768.005
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  • Exchangeable coalescents
  • Jean Bertoin, Université de Paris VI (Pierre et Marie Curie)
  • Book: Random Fragmentation and Coagulation Processes
  • Online publication: 07 December 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511617768.005
Available formats
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To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Exchangeable coalescents
  • Jean Bertoin, Université de Paris VI (Pierre et Marie Curie)
  • Book: Random Fragmentation and Coagulation Processes
  • Online publication: 07 December 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511617768.005
Available formats
×