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.

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

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

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

[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 | Zbl

[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

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

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

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

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

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

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

[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