Genealogies of regular exchangeable coalescents with applications to sampling
Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 3, p. 706-720

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.

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.

DOI : https://doi.org/10.1214/11-AIHP436
Classification:  60J25,  60F99,  92D25
Keywords: exchangeable coalescents, 𝛯-coalescent, 𝛬-coalescent, regularity, sampling formula, small-time asymptotics, coming down from infinity, martingale technique, random mutation rate
@article{AIHPB_2012__48_3_706_0,
     author = {Limic, Vlada},
     title = {Genealogies of regular exchangeable coalescents with applications to sampling},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {3},
     year = {2012},
     pages = {706-720},
     doi = {10.1214/11-AIHP436},
     zbl = {1271.92024},
     mrnumber = {2976560},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2012__48_3_706_0}
}
Limic, Vlada. Genealogies of regular exchangeable coalescents with applications to sampling. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 3, pp. 706-720. doi : 10.1214/11-AIHP436. http://www.numdam.org/item/AIHPB_2012__48_3_706_0/

[1] A.-L. Basdevant and C. Goldschmidt. Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent. Electron. J. Probab. 13 (2008) 486-512. | MR 2386740 | Zbl 1190.60006

[2] N. Berestycki. Recent Progress in Coalescent Theory. Ensaios matematicos [Mathematical Surveys] 16. Sociedade Brasileira de Matemática, Rio de Janeiro, 2009. | MR 2574323 | Zbl 1204.60002

[3] J. Berestycki, N. Berestycki and V. Limic. The 𝛬-coalescent speed of coming down from infinity. Ann. Probab. 38 (2010) 207-233. | MR 2599198 | Zbl 1247.60110

[4] J. Berestycki, N. Berestycki and V. Limic. Asymptotic sampling formulae and particle system representations for 𝛬-coalescents. Preprint. Available at http://www.cmi.univ-mrs.fr/~vlada/research.html, 2011.

[5] J. Berestycki, N. Berestycki and J. Schweinsberg. Beta-coalescents and continuous stable random trees. Ann. Probab. 35 (2007) 1835-1887. | MR 2349577 | Zbl 1129.60067

[6] J. Berestycki, N. Berestycki and J. Schweinsberg. Small-time behavior of beta-coalescents. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 214-238. | Numdam | MR 2446321 | Zbl 1214.60034

[7] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Univ. Press, Cambridge, 2006. | MR 2253162 | Zbl 1107.60002

[8] P. Donnelly and T. Kurtz. Particle representations for measure-valued population models. Ann. Probab. 27 (1999) 166-205. | MR 1681126 | Zbl 0956.60081

[9] M. Drmota, A. Iksanov, M. Möhle and U. Rösler. Asymptotic results concerning the total branch length of the Bolthausen-Sznitman coalescent. Stochastic Process. Appl. 117 (2007) 1404-1421. | MR 2353033 | Zbl 1129.60069

[10] R. Durrett. Probability: Theory and Examples, 3rd edition. Duxbury Advanced Series. Duxbury Press, Belmont, CA, 2004. | MR 2722836 | Zbl 0709.60002

[11] R. Durrett and J. Schweinsberg. A coalescent model for the effect of advantageous mutations on the genealogy of a population. Random partitions approximating the coalescence of lineages during a selective sweep. Stochastic Process. Appl. 115 (2005) 1628-1657. | MR 2165337 | Zbl 1082.92031

[12] W. J. Ewens. The sampling theory of selectively neutral alleles. Theor. Pop. Biol. 3 (1972) 87-112. | MR 325177 | Zbl 0245.92009

[13] A. Gnedin, B. Hansen and J. Pitman. Notes on the occupancy problem with infinitely many boxes: General asymptotics and power laws. Probab. Surv. 4 (2007) 146-171. | MR 2318403 | Zbl 1189.60050

[14] C. Foucart. Distinguished exchangeable coalescents and generalized Fleming-Viot processes with immigration. Preprint. Available at http://arxiv.org/abs/1006.0581, 2011. | MR 2848380 | Zbl pre05931191

[15] J. F. C. Kingman. The coalescent. Stochastic. Process. Appl. 13 (1982) 235-248. | MR 671034 | Zbl 0491.60076

[16] J. F. C. Kingman. On the genealogy of large populations. J. Appl. Probab. 19 (1982) 27-43. | MR 633178 | Zbl 0516.92011

[17] G. Li and D. Hedgecock. Genetic heterogeneity, detected by PCR SSCP, among samples of larval Pacific oysters (Crassostrea gigas) supports the hypothesis of large variance in reproductive success. Can. J. Fish. Aquat. Sci. 55 (1998) 1025-1033.

[18] V. Limic. On the speed of coming down from infinity for 𝛯-coalescent processes. Electron. J. Probab. 15 (2010) 217-240. | MR 2594877 | Zbl 1203.60111

[19] V. Limic. Coalescent processes and reinforced random walks: A guide through martingales and coupling. Habilitation thesis. Available at http://www.cmi.univ-mrs.fr/~vlada/research.html, 2011.

[20] M. Möhle. Coalescent processes without proper frequencies and applications to the two-parameter Poisson-Dirichlet coalescent. Preprint, 2009. | Zbl 1214.60037

[21] M. Möhle and S. Sagitov. A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29 (2001) 1547-1562. | MR 1880231 | Zbl 1013.92029

[22] E. Pardoux and M. Salamat. On the height and length of the Ancestral Recombination Graph. J. Appl. Probab. 46 (2009) 669-689. | MR 2560895 | Zbl 1176.60067

[23] J. Pitman. Coalescents with multiple collisions. Ann. Probab. 27 (1999) 1870-1902. | Zbl 0963.60079

[24] S. Sagitov. The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 (1999) 1116-1125. | Zbl 0962.92026

[25] J. Schweinsberg. Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5 (2000) 1-50. | Zbl 0959.60065

[26] J. Schweinsberg. The number of small blocks in exchangeable random partitions. ALEA 7 (2010) 217-242. | Zbl 1276.60011

[27] J. Schweinsberg and R. Durrett. Random partitions approximating the coalescence of lineages during a selective sweep. Ann. Appl. Probab. 15 (2005) 1591-1651. | Zbl 1073.92029