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2020 Kingman’s coalescent with erosion
Félix Foutel-Rodier, Amaury Lambert, Emmanuel Schertzer
Electron. J. Probab. 25: 1-33 (2020). DOI: 10.1214/20-EJP450

Abstract

Consider the Markov process taking values in the partitions of $\mathbb{N} $ such that each pair of blocks merges at rate one, and each integer is eroded, i.e., becomes a singleton block, at rate $d$. This is a special case of exchangeable fragmentation-coalescence process, called Kingman’s coalescent with erosion. We provide a new construction of the stationary distribution of this process as a sample from a standard flow of bridges. This allows us to give a representation of the asymptotic frequencies of this stationary distribution in terms of a sequence of independent diffusions. Moreover, we introduce a new process valued in the partitions of $\mathbb{Z} $ called Kingman’s coalescent with immigration, where pairs of blocks coalesce at rate one, and new blocks of size one immigrate according to a Poisson process of intensity $d$. By coupling Kingman’s coalescents with erosion and with immigration, we are able to show that the size of a block chosen uniformly at random from the stationary distribution of the restriction of Kingman’s coalescent with erosion to $\{1, \dots , n\}$ converges as $n\to \infty $ to the total progeny of a critical binary branching process.

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Félix Foutel-Rodier. Amaury Lambert. Emmanuel Schertzer. "Kingman’s coalescent with erosion." Electron. J. Probab. 25 1 - 33, 2020. https://doi.org/10.1214/20-EJP450

Information

Received: 12 July 2019; Accepted: 6 April 2020; Published: 2020
First available in Project Euclid: 5 May 2020

zbMATH: 07206395
MathSciNet: MR4095052
Digital Object Identifier: 10.1214/20-EJP450

Subjects:
Primary: 60G09
Secondary: 60G10, 60J25, 60J60, 60J80, 92D15

JOURNAL ARTICLE
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