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February 2019 Recovering the Brownian coalescent point process from the Kingman coalescent by conditional sampling
Amaury Lambert, Emmanuel Schertzer
Bernoulli 25(1): 148-173 (February 2019). DOI: 10.3150/17-BEJ971


We consider a continuous population whose dynamics is described by the standard stationary Fleming–Viot process, so that the genealogy of $n$ uniformly sampled individuals is distributed as the Kingman $n$-coalescent. In this note, we study some genealogical properties of this population when the sample is conditioned to fall entirely into a subpopulation with most recent common ancestor (MRCA) shorter than $\varepsilon$. First, using the comb representation of the total genealogy (Lambert and Uribe Bravo (P-Adic Numbers Ultrametric Anal. Appl. 9 (2017) 22–38)), we show that the genealogy of the descendance of the MRCA of the sample on the timescale $\varepsilon$ converges as $\varepsilon\to0$. The limit is the so-called Brownian coalescent point process (CPP) stopped at an independent Gamma random variable with parameter $n$, which can be seen as the genealogy at a large time of the total population of a rescaled critical birth–death process, biased by the $n$th power of its size. Second, we show that in this limit the coalescence times of the $n$ sampled individuals are i.i.d. uniform random variables in $(0,1)$. These results provide a coupling between two standard models for the genealogy of a random exchangeable population: the Kingman coalescent and the Brownian CPP.


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Amaury Lambert. Emmanuel Schertzer. "Recovering the Brownian coalescent point process from the Kingman coalescent by conditional sampling." Bernoulli 25 (1) 148 - 173, February 2019.


Received: 1 November 2016; Revised: 1 June 2017; Published: February 2019
First available in Project Euclid: 12 December 2018

zbMATH: 07007203
MathSciNet: MR3892315
Digital Object Identifier: 10.3150/17-BEJ971

Keywords: Coalescent point process , conditional sampling , flows of bridges , Kingman coalescent , small time behavior

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 1 • February 2019
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