Genealogies of regular exchangeable coalescents with applications to sampling

Vlada Limic

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 3, page 706-720
  • ISSN: 0246-0203

Abstract

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This article considers a model of genealogy corresponding to a regular exchangeable coalescent (also known as 𝛯 -coalescent) started from a large finite configuration, and undergoing neutral mutations. Asymptotic expressions for the number of active lineages were obtained by the author in a previous work. Analogous results for the number of active mutation-free lineages and the combined lineage lengths are derived using the same martingale-based technique. They are given in terms of convergence in probability, while extensions to convergence in moments and convergence almost surely are discussed. The above mentioned results have direct consequences on the sampling theory in the 𝛯 -coalescent setting. In particular, the regular 𝛯 -coalescents that come down from infinity (i.e., with locally finite genealogies) have an asymptotically equal number of families under the corresponding infinite alleles and infinite sites models. In special cases, quantitative asymptotic formulae for the number of families that contain a fixed number of individuals can be given.

How to cite

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Limic, Vlada. "Genealogies of regular exchangeable coalescents with applications to sampling." Annales de l'I.H.P. Probabilités et statistiques 48.3 (2012): 706-720. <http://eudml.org/doc/272008>.

@article{Limic2012,
abstract = {This article considers a model of genealogy corresponding to a regular exchangeable coalescent (also known as $\varXi $-coalescent) started from a large finite configuration, and undergoing neutral mutations. Asymptotic expressions for the number of active lineages were obtained by the author in a previous work. Analogous results for the number of active mutation-free lineages and the combined lineage lengths are derived using the same martingale-based technique. They are given in terms of convergence in probability, while extensions to convergence in moments and convergence almost surely are discussed. The above mentioned results have direct consequences on the sampling theory in the $\varXi $-coalescent setting. In particular, the regular $\varXi $-coalescents that come down from infinity (i.e., with locally finite genealogies) have an asymptotically equal number of families under the corresponding infinite alleles and infinite sites models. In special cases, quantitative asymptotic formulae for the number of families that contain a fixed number of individuals can be given.},
author = {Limic, Vlada},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {exchangeable coalescents; $\varXi $-coalescent; $\varLambda $-coalescent; regularity; sampling formula; small-time asymptotics; coming down from infinity; martingale technique; random mutation rate; exchangeable coalescents; -coalescent; -coalescent; regularity; sampling formula; small-time asymptotics; coming down from infinity; martingale techniques; random mutation rate},
language = {eng},
number = {3},
pages = {706-720},
publisher = {Gauthier-Villars},
title = {Genealogies of regular exchangeable coalescents with applications to sampling},
url = {http://eudml.org/doc/272008},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Limic, Vlada
TI - Genealogies of regular exchangeable coalescents with applications to sampling
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 3
SP - 706
EP - 720
AB - This article considers a model of genealogy corresponding to a regular exchangeable coalescent (also known as $\varXi $-coalescent) started from a large finite configuration, and undergoing neutral mutations. Asymptotic expressions for the number of active lineages were obtained by the author in a previous work. Analogous results for the number of active mutation-free lineages and the combined lineage lengths are derived using the same martingale-based technique. They are given in terms of convergence in probability, while extensions to convergence in moments and convergence almost surely are discussed. The above mentioned results have direct consequences on the sampling theory in the $\varXi $-coalescent setting. In particular, the regular $\varXi $-coalescents that come down from infinity (i.e., with locally finite genealogies) have an asymptotically equal number of families under the corresponding infinite alleles and infinite sites models. In special cases, quantitative asymptotic formulae for the number of families that contain a fixed number of individuals can be given.
LA - eng
KW - exchangeable coalescents; $\varXi $-coalescent; $\varLambda $-coalescent; regularity; sampling formula; small-time asymptotics; coming down from infinity; martingale technique; random mutation rate; exchangeable coalescents; -coalescent; -coalescent; regularity; sampling formula; small-time asymptotics; coming down from infinity; martingale techniques; random mutation rate
UR - http://eudml.org/doc/272008
ER -

References

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