Recovering the Brownian coalescent point process from the Kingman coalescent by conditional sampling

摘要

We consider a continuous population whose dynamics is described by thestandard stationary Fleming-Viot process, so that the genealogy of $n$uniformly sampled individuals is distributed as the Kingman $n$-coalescent. Inthis note, we study some genealogical properties of this population when thesample is conditioned to fall entirely into a subpopulation with most recentcommon ancestor (MRCA) shorter than $arepsilon$. First, using the combrepresentation of the total genealogy (Lambert & Uribe Bravo 2016), we showthat the genealogy of the descendance of the MRCA of the sample on thetimescale $arepsilon$ converges as $arepsilono 0$. The limit is theso-called Brownian coalescent point process (CPP) stopped at an independentGamma random variable with parameter $n$, which can be seen as the genealogy ata large time of the total population of a rescaled critical birth-deathprocess, biased by the $n$-th power of its size. Secondly, we show that in thislimit the coalescence times of the $n$ sampled individuals are i.i.d. uniformrandom variables in $(0,1)$. These results provide a coupling between twostandard models for the genealogy of a random exchangeable population: theKingman coalescent and the Brownian CPP.