Genealogies of regular exchangeable coalescents with applications to sampling

Vlada Limic

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 3, page 706-720
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Limic, Vlada. "Genealogies of regular exchangeable coalescents with applications to sampling." Annales de l'I.H.P. Probabilités et statistiques 48.3 (2012): 706-720. <http://eudml.org/doc/272008>.

@article{Limic2012,
abstract = {This article considers a model of genealogy corresponding to a regular exchangeable coalescent (also known as $\varXi $-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 $\varXi $-coalescent setting. In particular, the regular $\varXi $-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.},
author = {Limic, Vlada},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {exchangeable coalescents; $\varXi $-coalescent; $\varLambda $-coalescent; regularity; sampling formula; small-time asymptotics; coming down from infinity; martingale technique; random mutation rate; exchangeable coalescents; -coalescent; -coalescent; regularity; sampling formula; small-time asymptotics; coming down from infinity; martingale techniques; random mutation rate},
language = {eng},
number = {3},
pages = {706-720},
publisher = {Gauthier-Villars},
title = {Genealogies of regular exchangeable coalescents with applications to sampling},
url = {http://eudml.org/doc/272008},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Limic, Vlada
TI - Genealogies of regular exchangeable coalescents with applications to sampling
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 3
SP - 706
EP - 720
AB - This article considers a model of genealogy corresponding to a regular exchangeable coalescent (also known as $\varXi $-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 $\varXi $-coalescent setting. In particular, the regular $\varXi $-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.
LA - eng
KW - exchangeable coalescents; $\varXi $-coalescent; $\varLambda $-coalescent; regularity; sampling formula; small-time asymptotics; coming down from infinity; martingale technique; random mutation rate; exchangeable coalescents; -coalescent; -coalescent; regularity; sampling formula; small-time asymptotics; coming down from infinity; martingale techniques; random mutation rate
UR - http://eudml.org/doc/272008
ER -

References

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  1. [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. Zbl1190.60006MR2386740
  2. [2] N. Berestycki. Recent Progress in Coalescent Theory. Ensaios matematicos [Mathematical Surveys] 16. Sociedade Brasileira de Matemática, Rio de Janeiro, 2009. Zbl1204.60002MR2574323
  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. Asymptotic sampling formulae and particle system representations for 𝛬 -coalescents. Preprint. Available at http://www.cmi.univ-mrs.fr/~vlada/research.html, 2011. Zbl1321.60146
  5. [5] J. Berestycki, N. Berestycki and J. Schweinsberg. Beta-coalescents and continuous stable random trees. Ann. Probab.35 (2007) 1835–1887. Zbl1129.60067MR2349577
  6. [6] J. Berestycki, N. Berestycki and J. Schweinsberg. Small-time behavior of beta-coalescents. Ann. Inst. H. Poincaré Probab. Statist.44 (2008) 214–238. Zbl1214.60034MR2446321
  7. [7] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Univ. Press, Cambridge, 2006. Zbl1107.60002MR2253162
  8. [8] P. Donnelly and T. Kurtz. Particle representations for measure-valued population models. Ann. Probab.27 (1999) 166–205. Zbl0956.60081MR1681126
  9. [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. Zbl1129.60069MR2353033
  10. [10] R. Durrett. Probability: Theory and Examples, 3 rd edition. Duxbury Advanced Series. Duxbury Press, Belmont, CA, 2004. Zbl0709.60002MR2722836
  11. [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. Zbl1082.92031MR2165337
  12. [12] W. J. Ewens. The sampling theory of selectively neutral alleles. Theor. Pop. Biol.3 (1972) 87–112. Zbl0245.92009MR325177
  13. [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. Zbl1189.60050MR2318403
  14. [14] C. Foucart. Distinguished exchangeable coalescents and generalized Fleming–Viot processes with immigration. Preprint. Available at http://arxiv.org/abs/1006.0581, 2011. Zbl1300.60086MR2848380
  15. [15] J. F. C. Kingman. The coalescent. Stochastic. Process. Appl.13 (1982) 235–248. Zbl0491.60076MR671034
  16. [16] J. F. C. Kingman. On the genealogy of large populations. J. Appl. Probab.19 (1982) 27–43. Zbl0516.92011MR633178
  17. [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. [18] V. Limic. On the speed of coming down from infinity for 𝛯 -coalescent processes. Electron. J. Probab.15 (2010) 217–240. Zbl1203.60111MR2594877
  19. [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. [20] M. Möhle. Coalescent processes without proper frequencies and applications to the two-parameter Poisson–Dirichlet coalescent. Preprint, 2009. Zbl1214.60037
  21. [21] M. Möhle and S. Sagitov. A classification of coalescent processes for haploid exchangeable population models. Ann. Probab.29 (2001) 1547–1562. Zbl1013.92029MR1880231
  22. [22] E. Pardoux and M. Salamat. On the height and length of the Ancestral Recombination Graph. J. Appl. Probab.46 (2009) 669–689. Zbl1176.60067MR2560895
  23. [23] J. Pitman. Coalescents with multiple collisions. Ann. Probab.27 (1999) 1870–1902. Zbl0963.60079
  24. [24] S. Sagitov. The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab.36 (1999) 1116–1125. Zbl0962.92026
  25. [25] J. Schweinsberg. Coalescents with simultaneous multiple collisions. Electron. J. Probab.5 (2000) 1–50. Zbl0959.60065
  26. [26] J. Schweinsberg. The number of small blocks in exchangeable random partitions. ALEA7 (2010) 217–242. Zbl1276.60011
  27. [27] J. Schweinsberg and R. Durrett. Random partitions approximating the coalescence of lineages during a selective sweep. Ann. Appl. Probab.15 (2005) 1591–1651. Zbl1073.92029

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