Asymptotic sampling formulae for $\varLambda $-coalescents

Berestycki, Julien; Berestycki, Nathanaël; Limic, Vlada

Asymptotic sampling formulae for

𝛬

-coalescents

Berestycki, Julien ; Berestycki, Nathanaël ; Limic, Vlada

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014), p. 715-731 / Harvested from Numdam

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Résumé
Abstract

Nous présentons une méthode robuste qui permet de traduire des informations sur la vitesse de descente de l’infini d’un arbre généalogique en formules d’échantillonnages pour la population sous-jacente. Nous appliquons cette méthode au cas où la génélaogie est donnée par un $𝛬$ -coalescent. Nous en déduisons une formule exacte pour le comportement asymptotique du spectre des fréquences alléliques et du nombre de sites de ségrégation, lorsque la taille de l’échantillon tend vers l’infini. Certains de ces résultats sont valides dans le cas général où le coalescent descend de l’infini, tandis que d’autres plus précis sont obtenus sous une hypothèse de variation régulière. Dans ce cas nous obtenons également des résultats, dont l’intérêt dépasse ce contexte, sur le temps auquel une mutation choisie uniformément au hasard est apparue. Il apparaît que cette quantité connaît une transition de phase autour de la valeur $α = 3 / 2$ , où $α$ est l’exposant de variation régulière.

We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a $𝛬$ -coalescent. This allows us to derive an exact formula for the asymptotic behavior of the site and allele frequency spectrum and the number of segregating sites, as the sample size tends to $\infty$ . Some of our results hold in the case of a general $𝛬$ -coalescent that comes down from infinity, but we obtain more precise information under a regular variation assumption. In this case, we obtain results of independent interest for the time at which a mutation uniformly chosen at random was generated. This exhibits a phase transition at $α = 3 / 2$ , where $α \in (1, 2)$ is the exponent of regular variation.

Publié le : 2014-01-01

MR 3224287 | Zbl 06340406

DOI : https://doi.org/10.1214/13-AIHP546
Classification: 60J25, 60F99, 92D25

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     author = {Berestycki, Julien and Berestycki, Nathana\"el and Limic, Vlada},
     title = {Asymptotic sampling formulae for $\varLambda $-coalescents},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {50},
     year = {2014},
     pages = {715-731},
     doi = {10.1214/13-AIHP546},
     mrnumber = {3224287},
     zbl = {06340406},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2014__50_3_715_0}
}

Berestycki, Julien; Berestycki, Nathanaël; Limic, Vlada. Asymptotic sampling formulae for $\varLambda $-coalescents. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) pp. 715-731. doi : 10.1214/13-AIHP546. http://gdmltest.u-ga.fr/item/AIHPB_2014__50_3_715_0/

Bibliographie

[1] D. J. Aldous. Exchangeability and related topics. In École d'Eté de Probabilités de Saint-Flour XIII - 1983. Lecture Notes Math. 1117. Springer, Berlin, 1985. | MR 883646 | Zbl 0562.60042

[2] 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

[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. A small-time coupling between $𝛬$ -coalescents and branching processes. Preprint, 2012. | Zbl pre06291796

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

[6] 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

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

[8] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics. Cambridge Univ. Press, Cambridge, 2006. | MR 2253162 | Zbl 1107.60002

[9] J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes III: Limit theorems. Illinois J. Math. 50 (2006) 147-181. | MR 2247827 | Zbl 1110.60026

[10] J.-S. Dhersin, F. Freund, A. Siri-Jegousse and L. Yuan. On the length of an external branch in the Beta-coalescent, 2012. Available at arXiv:1201.3983. | MR 3027896 | Zbl 1281.60069

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

[12] R. Durrett. Probability Models for DNA Sequence Evolution. Springer, Berlin, 2002. | MR 1903526 | Zbl pre05280644

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

[14] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley, New York, 1971. | MR 270403 | Zbl 0219.60003

[15] N. Freeman. The number of non-singleton blocks in Lambda-coalescents with dust, 2011. Available at arXiv:1111.1660.

[16] 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

[17] G. Kersting. The asymptotic distribution of the length of Beta-coalescent trees. Ann. Appl. Probab. 22 (2012) 2086-2107. | MR 3025690 | Zbl 1251.92034

[18] M. Kimura. The number of heterozygous nucleotide sites maintained in a finite population due to steady flux of mutations. Genetics 61 (1969) 893-903.

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

[20] 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

[21] V. Limic. Genealogies of regular exchangeable coalescents with applications to sampling. Ann. Inst. Henri Poincaré Probab. Statist. 48 (2012) 706-720. | Numdam | MR 2976560 | Zbl 1271.92024

[22] V. Limic. Processus de Coalescence et Marches Aléatoires Renforcées : Un guide à travers martingales et couplage. Habilitation thesis (in French and English), 2011. Available at http://www.latp.univ-mrs.fr/~vlada/habi.html.

[23] M. Möhle. On the number of segregating sites for populations with large family sizes. Adv. in Appl. Probab. 38 (2006) 750-767. | MR 2256876 | Zbl 1112.92046

[24] 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

[25] J. Pitman. Coalescents with multiple collisions. Ann. Probab. 27 (1999) 1870-1902. | MR 1742892 | Zbl 0963.60079

[26] J. Pitman. Combinatorial stochastic processes. In École d'Eté de Probabilités de Saint-Flour XXXII - 2002. Lecture Notes Math. 1875. Springer, Berlin, 2006. | MR 2245368 | Zbl 1103.60004

[27] S. Sagitov. The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 (1999) 1116-1125. | MR 1742154 | Zbl 0962.92026

[28] J. Schweinsberg. A necessary and sufficient condition for the $𝛬$ -coalescent to come down from infinity. Electron. Commun. Probab. 5 (2000) 1-11. | MR 1736720 | Zbl 0953.60072

[29] J. Schweinsberg. The number of small blocks in exchangeable random partitions. ALEA Lat. Am. J. Probab. Math. Stat. 7 (2010) 217-242. | MR 2672786 | Zbl 1276.60011