The coalescent point process of branching trees and spine decompositions at the first survivor

Abstract : We define a quasistationary monotone labelling of Bienaymé--Galton--Watson (BGW) genealogies. The genealogy of the current generation backwards in time is uniquely determined by the coalescent point process $(A_i;i\ge 1)$, where $A_i$ is the coalescence time between individuals $i$ and $i+1$. There is a Markov process of point measures $(B_i;i\ge 1)$ keeping track of more ancestral relationships, such that $A_i$ is also the first point mass of $B_i$. We further give an inhomogeneous spine decomposition ending at the first surviving particle of generation $h$ in a plane BGW tree conditioned to survive $h$ generations. The decomposition involves a point measure $\rho$ storing the number of subtrees on the right-hand side of the spine. Under appropriate conditions, we prove convergence of this point measure to a point measure on $\RR_+$ associated with the limiting continuous-state branching process. We prove the associated invariance principle for the coalescent point process, after we discretize the limiting continuous-state population by considering only points with coalescence times greater than $\vareps$. The limiting coalescent point process is the sequence of depths greater than $\vareps$ of the excursions of the height process below some fixed level. In the diffusion case, there are no multiple ancestries and (it is known that) the coalescent point process is a Poisson point process with an explicit intensity measure. The question of characterizing this point process in the presence of multiple repeats as $\vareps\to 0$ remains open.
Type de document :
Pré-publication, Document de travail
40 pages, 2 figures. 2011
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https://hal.archives-ouvertes.fr/hal-00555593
Contributeur : Amaury Lambert <>
Soumis le : jeudi 13 janvier 2011 - 21:19:53
Dernière modification le : mercredi 12 octobre 2016 - 01:03:56

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  • HAL Id : hal-00555593, version 1
  • ARXIV : 1101.2241

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UPMC | PMA | INSMI | USPC

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Amaury Lambert, Lea Popovic. The coalescent point process of branching trees and spine decompositions at the first survivor. 40 pages, 2 figures. 2011. <hal-00555593>

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