Genealogies of regular exchangeable coalescents with applications to sampling

Limic, Vlada

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 3, pp. 706-720.

Cet article considère un modèle de généalogie qui correspond à un processus de coalescence échangeable régulier (appelé aussi un $𝛯$-coalescent) démarré d’une configuration à taille finie et grande, et subissant des mutations neutres. Des expressions asymptotiques pour le nombre de lignées actives ont été obtenues par l’auteur dans un travail précédent. Des résultats analogues pour le nombre de lignées actives et la longueur totale des lignées sont dérivés par les mêmes techniques martingales. Ils sont donnés en terme de la convergence en probabilité, pendant que des extensions à la convergence au sens des moments et la convergence presque sûre sont examinées. Ces résultats ont des conséquences directes sur la théorie d’échantillonnage dans le cadre de $𝛯$-coalescence. En particulier, les $𝛯$-coalescents réguliers qui descendent de l’infini (c.-à-d. qui ont des généalogies localement finies) ont des nombres de familles égaux au sens asymptotique sous le modèle d’allèles infinies et le modèle de site infinis. Dans des cas particuliers, on peut ainsi dériver des formules asymptotiques quantitatives pour le nombre de familles contenant un nombre fixe d’individus.

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.

Classification : 60J25, 60F99, 92D25
Mots clés : exchangeable coalescents, $\varXi $-coalescent, $\varLambda $-coalescent, regularity, sampling formula, small-time asymptotics, coming down from infinity, martingale technique, random mutation rate

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