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February 2022 The symmetric coalescent and Wright–Fisher models with bottlenecks
Adrián González Casanova, Verónica Miró Pina, Arno Siri-Jégousse
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Ann. Appl. Probab. 32(1): 235-268 (February 2022). DOI: 10.1214/21-AAP1676

Abstract

We define a new class of Ξ-coalescents characterized by a possibly infinite measure over the nonnegative integers. We call them symmetric coalescents since they are the unique family of exchangeable coalescents satisfying a symmetry property on their coagulation rates: they are invariant under any transformation that consists of moving one element from one block to another without changing the total number of blocks. We illustrate the diversity of behaviors of this family of processes by introducing and studying a one parameter subclass, the (β,S)-coalescents. We also embed this family in a larger class of Ξ-coalescents arising as the limit genealogies of Wright–Fisher models with bottlenecks. Some convergence results rely on a new Skorokhod type metric, that induces the Meyer–Zheng topology, which allows us to study the scaling limit of non-Markovian processes using standard techniques.

Funding Statement

AGC was supported by CONACyT Grant A1-S-14615, VMP by DGAPA-UNAM postdoctoral program and ASJ by CONACyT Grant CB-2014/243068.

Acknowledgments

The authors thank three anonymous referees for their comments which helped improve this paper. ACG thanks Jochen Blath for helpful discussions.

Citation

Download Citation

Adrián González Casanova. Verónica Miró Pina. Arno Siri-Jégousse. "The symmetric coalescent and Wright–Fisher models with bottlenecks." Ann. Appl. Probab. 32 (1) 235 - 268, February 2022. https://doi.org/10.1214/21-AAP1676

Information

Received: 1 June 2019; Revised: 1 September 2020; Published: February 2022
First available in Project Euclid: 27 February 2022

Digital Object Identifier: 10.1214/21-AAP1676

Subjects:
Primary: 60K35
Secondary: 60J80

Keywords: bottlenecks , coalescent processes , convergence in measure , moment duality , Population genetics , Skorokhod topology , symmetric coalescent , Ξ-coalescent

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.32 • No. 1 • February 2022
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