Asymptotic sampling formulae for 𝛬 -coalescents

Julien Berestycki; Nathanaël Berestycki; Vlada Limic

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 3, page 715-731
  • ISSN: 0246-0203

Abstract

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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 ∞ . 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 ∈ ( 1 , 2 ) is the exponent of regular variation.

How to cite

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Berestycki, Julien, Berestycki, Nathanaël, and Limic, Vlada. "Asymptotic sampling formulae for $\varLambda $-coalescents." Annales de l'I.H.P. Probabilités et statistiques 50.3 (2014): 715-731. <http://eudml.org/doc/272053>.

@article{Berestycki2014,
abstract = {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 $\varLambda $-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 $\varLambda $-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 $\alpha =3/2$, where $\alpha \in (1,2)$ is the exponent of regular variation.},
author = {Berestycki, Julien, Berestycki, Nathanaël, Limic, Vlada},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {$\varLambda $-coalescents; speed of coming down from infinity; exchangeable coalescents; sampling formulae; infinite allele model; genetic variation; -coalescents; exchangeable coalescents; sampling formulae; infinite allele model; population dynamics; genetic variation},
language = {eng},
number = {3},
pages = {715-731},
publisher = {Gauthier-Villars},
title = {Asymptotic sampling formulae for $\varLambda $-coalescents},
url = {http://eudml.org/doc/272053},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Berestycki, Julien
AU - Berestycki, Nathanaël
AU - Limic, Vlada
TI - Asymptotic sampling formulae for $\varLambda $-coalescents
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 3
SP - 715
EP - 731
AB - 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 $\varLambda $-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 $\varLambda $-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 $\alpha =3/2$, where $\alpha \in (1,2)$ is the exponent of regular variation.
LA - eng
KW - $\varLambda $-coalescents; speed of coming down from infinity; exchangeable coalescents; sampling formulae; infinite allele model; genetic variation; -coalescents; exchangeable coalescents; sampling formulae; infinite allele model; population dynamics; genetic variation
UR - http://eudml.org/doc/272053
ER -

References

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  1. [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. Zbl0562.60042MR883646
  2. [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. Zbl1190.60006MR2386740
  3. [3] J. Berestycki, N. Berestycki and V. Limic. The 𝛬 -coalescent speed of coming down from infinity. Ann. Probab.38 (2010) 207–233. Zbl1247.60110MR2599198
  4. [4] J. Berestycki, N. Berestycki and V. Limic. A small-time coupling between 𝛬 -coalescents and branching processes. Preprint, 2012. Zbl1303.60066
  5. [5] J. Berestycki, N. Berestycki and J. Schweinsberg. Small-time behavior of beta-coalescents. Ann. Inst. Henri Poincaré Probab. Stat.44 (2008) 214–238. Zbl1214.60034MR2446321
  6. [6] J. Berestycki, N. Berestycki and J. Schweinsberg. Beta-coalescents and continuous stable random trees. Ann. Probab.35 (2007) 1835–1887. Zbl1129.60067MR2349577
  7. [7] N. Berestycki. Recent Progress in Coalescent Theory. Ensaios Matematicos 16. Sociedade Brasileira de Matemática, Rio de Janeiro, 2009. Zbl1204.60002MR2574323
  8. [8] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Studies in Advanced Mathematics. Cambridge Univ. Press, Cambridge, 2006. Zbl1107.60002MR2253162
  9. [9] J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes III: Limit theorems. Illinois J. Math.50 (2006) 147–181. Zbl1110.60026MR2247827
  10. [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. Zbl1281.60069MR3027896
  11. [11] P. Donnelly and T. Kurtz. Particle representations for measure-valued population models. Ann. Probab.27 (1999) 166–205. Zbl0956.60081MR1681126
  12. [12] R. Durrett. Probability Models for DNA Sequence Evolution. Springer, Berlin, 2002. Zbl1311.92007MR1903526
  13. [13] W. J. Ewens. The sampling theory of selectively neutral alleles. Theor. Pop. Biol.3 (1972) 87–112. Zbl0245.92009MR325177
  14. [14] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley, New York, 1971. Zbl0219.60003MR270403
  15. [15] N. Freeman. The number of non-singleton blocks in Lambda-coalescents with dust, 2011. Available at arXiv:1111.1660. 
  16. [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. Zbl1189.60050MR2318403
  17. [17] G. Kersting. The asymptotic distribution of the length of Beta-coalescent trees. Ann. Appl. Probab.22 (2012) 2086–2107. Zbl1251.92034MR3025690
  18. [18] M. Kimura. The number of heterozygous nucleotide sites maintained in a finite population due to steady flux of mutations. Genetics61 (1969) 893–903. 
  19. [19] J. F. C. Kingman. On the genealogies of large populations. J. Appl. Probab.19 (1982) 27–43. Zbl0516.92011MR633178
  20. [20] V. Limic. On the speed of coming down from infinity for 𝛯 -coalescent processes. Electron. J. Probab.15 (2010) 217–240. Zbl1203.60111MR2594877
  21. [21] V. Limic. Genealogies of regular exchangeable coalescents with applications to sampling. Ann. Inst. Henri Poincaré Probab. Statist.48 (2012) 706–720. Zbl1271.92024MR2976560
  22. [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. [23] M. Möhle. On the number of segregating sites for populations with large family sizes. Adv. in Appl. Probab.38 (2006) 750–767. Zbl1112.92046MR2256876
  24. [24] M. Möhle and S. Sagitov. A classification of coalescent processes for haploid exchangeable population models. Ann Probab.29 (2001) 1547–1562. Zbl1013.92029MR1880231
  25. [25] J. Pitman. Coalescents with multiple collisions. Ann. Probab.27 (1999) 1870–1902. Zbl0963.60079MR1742892
  26. [26] J. Pitman. Combinatorial stochastic processes. In École d’Eté de Probabilités de Saint-Flour XXXII – 2002. Lecture Notes Math. 1875. Springer, Berlin, 2006. Zbl1103.60004MR2245368
  27. [27] S. Sagitov. The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab.36 (1999) 1116–1125. Zbl0962.92026MR1742154
  28. [28] J. Schweinsberg. A necessary and sufficient condition for the 𝛬 -coalescent to come down from infinity. Electron. Commun. Probab.5 (2000) 1–11. Zbl0953.60072MR1736720
  29. [29] J. Schweinsberg. The number of small blocks in exchangeable random partitions. ALEA Lat. Am. J. Probab. Math. Stat.7 (2010) 217–242. Zbl1276.60011MR2672786

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