Small-time behavior of beta coalescents

Julien Berestycki; Nathanaël Berestycki; Jason Schweinsberg

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 2, page 214-238
  • ISSN: 0246-0203

Abstract

top
For a finite measure Λ on [0, 1], the Λ-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate ∫01xk−2(1−x)b−kΛ(dx). It has recently been shown that if 1<α<2, the Λ-coalescent in which Λ is the Beta (2−α, α) distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an α-stable branching mechanism. Here we use facts about CSBPs to establish new results about the small-time asymptotics of beta coalescents. We prove an a.s. limit theorem for the number of blocks at small times, and we establish results about the sizes of the blocks. We also calculate the Hausdorff and packing dimensions of a metric space associated with the beta coalescents, and we find the sum of the lengths of the branches in the coalescent tree, both of which are determined by the behavior of coalescents at small times. We extend most of these results to other Λ-coalescents for which Λ has the same asymptotic behavior near zero as the Beta (2−α, α) distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study small-time properties of Λ-coalescents.

How to cite

top

Berestycki, Julien, Berestycki, Nathanaël, and Schweinsberg, Jason. "Small-time behavior of beta coalescents." Annales de l'I.H.P. Probabilités et statistiques 44.2 (2008): 214-238. <http://eudml.org/doc/77967>.

@article{Berestycki2008,
abstract = {For a finite measure Λ on [0, 1], the Λ-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate ∫01xk−2(1−x)b−kΛ(dx). It has recently been shown that if 1&lt;α&lt;2, the Λ-coalescent in which Λ is the Beta (2−α, α) distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an α-stable branching mechanism. Here we use facts about CSBPs to establish new results about the small-time asymptotics of beta coalescents. We prove an a.s. limit theorem for the number of blocks at small times, and we establish results about the sizes of the blocks. We also calculate the Hausdorff and packing dimensions of a metric space associated with the beta coalescents, and we find the sum of the lengths of the branches in the coalescent tree, both of which are determined by the behavior of coalescents at small times. We extend most of these results to other Λ-coalescents for which Λ has the same asymptotic behavior near zero as the Beta (2−α, α) distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study small-time properties of Λ-coalescents.},
author = {Berestycki, Julien, Berestycki, Nathanaël, Schweinsberg, Jason},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {coalescence; continuous-state branching process; coalescent with multiple mergers},
language = {eng},
number = {2},
pages = {214-238},
publisher = {Gauthier-Villars},
title = {Small-time behavior of beta coalescents},
url = {http://eudml.org/doc/77967},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Berestycki, Julien
AU - Berestycki, Nathanaël
AU - Schweinsberg, Jason
TI - Small-time behavior of beta coalescents
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 2
SP - 214
EP - 238
AB - For a finite measure Λ on [0, 1], the Λ-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate ∫01xk−2(1−x)b−kΛ(dx). It has recently been shown that if 1&lt;α&lt;2, the Λ-coalescent in which Λ is the Beta (2−α, α) distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an α-stable branching mechanism. Here we use facts about CSBPs to establish new results about the small-time asymptotics of beta coalescents. We prove an a.s. limit theorem for the number of blocks at small times, and we establish results about the sizes of the blocks. We also calculate the Hausdorff and packing dimensions of a metric space associated with the beta coalescents, and we find the sum of the lengths of the branches in the coalescent tree, both of which are determined by the behavior of coalescents at small times. We extend most of these results to other Λ-coalescents for which Λ has the same asymptotic behavior near zero as the Beta (2−α, α) distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study small-time properties of Λ-coalescents.
LA - eng
KW - coalescence; continuous-state branching process; coalescent with multiple mergers
UR - http://eudml.org/doc/77967
ER -

References

top
  1. [1] D. J. Aldous. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5 (1999) 3–48. Zbl0930.60096MR1673235
  2. [2] A.-L. Basdevant. Ruelle’s probability cascades seen as a fragmentation process. Markov Process. Related Fields 12 (2006) 447–474. Zbl1113.60075MR2246260
  3. [3] J. Berestycki, N. Berestycki and J. Schweinsberg. Beta-coalescents and continuous stable random trees. Ann. Probab. 35 (2007) 1835–1887. Zbl1129.60067MR2349577
  4. [4] J. Bertoin. Lévy Processes. Cambridge University Press, Cambridge, 1996. Zbl0861.60003MR1406564
  5. [5] J. Bertoin. Random Coagulation and Fragmentation Processes. Cambridge University Press, Cambridge, 2006. Zbl1107.60002MR2253162
  6. [6] J. Bertoin and J.-F. Le Gall. The Bolthausen–Sznitman coalescent and the genealogy of continuous-state branching processes. Probab. Theory Related Fields 117 (2000) 249–266. Zbl0963.60086MR1771663
  7. [7] J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 (2003) 261–288. Zbl1023.92018MR1990057
  8. [8] 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
  9. [9] J. Bertoin and J. Pitman. Two coalescents derived from the ranges of stable subordinators. Electron. J. Probab. 5 (2000) 1–17. Zbl0949.60034MR1768841
  10. [10] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge University Press, Cambridge, 1987. Zbl0617.26001MR898871
  11. [11] M. Birkner, J. Blath, M. Capaldo, A. Etheridge, M. Möhle, J. Schweinsberg and A. Wakolbinger. Alpha-stable branching and beta-coalescents. Electron. J. Probab. 10 (2005) 303–325. Zbl1066.60072MR2120246
  12. [12] E. Bolthausen and A.-S. Sznitman. On Ruelle’s probability cascades and an abstract cavity method. Comm. Math. Phys. 197 (1998) 247–276. Zbl0927.60071MR1652734
  13. [13] P. Donnelly, S. N. Evans, K. Fleischmann, T. G. Kurtz and X. Zhou. Continuum-sites stepping-stone models, coalescing exchangeable partitions, and random trees. Ann. Probab. 28 (2000) 1063–1110. Zbl1023.60082MR1797304
  14. [14] P. Donnelly and T. G. Kurtz. Particle representations for measure-valued population models. Ann. Probab. 27 (1999) 166–205. Zbl0956.60081MR1681126
  15. [15] R. M. Dudley. Real Analysis and Probability. Wadsworth and Brooks/Cole, Pacific Grove, CA, 1989. Zbl0686.60001MR982264
  16. [16] T. Duquesne. A limit theorem for the contour process of conditioned Galton–Watson trees. Ann. Probab. 31 (2003) 996–1027. Zbl1025.60017MR1964956
  17. [17] T. Duquesne and J.-F. Le Gall. Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 (2005) 553–603. Zbl1070.60076MR2147221
  18. [18] R. Durrett and J. Schweinsberg. A coalescent model for the effect of advantageous mutations on the genealogy of a population. Stochastic Process. Appl. 115 (2005) 1628–1657. Zbl1082.92031MR2165337
  19. [19] N. El Karoui and S. Roelly. Propriétés de martingales, explosion et représentation de Lévy–Khintchine d’une classe de processus de branchement à valeurs mesures. Stochastic Process. Appl. 38 (1991) 239–266. Zbl0743.60081MR1119983
  20. [20] S. N. Evans. Kingman’s coalescent as a random metric space. In Stochastic Models: A Conference in Honour of Professor Donald A. Dawson (L. G. Gorostiza and B. G. Ivanoff, Eds). Canadian Mathematical Society/American Mathematical Society, 2000. Zbl0955.60010MR1765005
  21. [21] K. Falconer. Fractal Geometry: Mathematical Foundations and Applications, 2nd edition. Wiley, Hoboken, NJ, 2003. Zbl1060.28005MR2118797
  22. [22] C. Goldschmidt and J. Martin. Random recursive trees and the Bolthausen–Sznitman coalescent. Electron. J. Probab. 10 (2005) 718–745. Zbl1109.60060MR2164028
  23. [23] J. Kesten, P. Ney and F. Spitzer. The Galton–Watson process with mean one and finite variance. Theory Probab. Appl. 11 (1966) 513–540. Zbl0158.35202MR207052
  24. [24] J. F. C. Kingman. The representation of partition structures. J. London Math. Soc. 18 (1978) 374–380. Zbl0415.92009MR509954
  25. [25] J. F. C. Kingman. The coalescent. Stochastic Process. Appl. 13 (1982) 235–248. Zbl0491.60076MR671034
  26. [26] J. Lamperti. The limit of a sequence of branching processes. Z. Wahrsch. Verw. Gebiete 7 (1967) 271–288. Zbl0154.42603MR217893
  27. [27] J. Lamperti. Continuous state branching processes. Bull. Amer. Math. Soc. 73 (1967) 382–386. Zbl0173.20103MR208685
  28. [28] M. Möhle. On the number of segregating sites for populations with large family sizes. Adv. in Appl. Probab. 38 (2006) 750–767. Zbl1112.92046MR2256876
  29. [29] M. Möhle and S. Sagitov. A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29 (2001) 1547–1562. Zbl1013.92029MR1880231
  30. [30] J. Pitman. Coalescents with multiple collisions. Ann. Probab. 27 (1999) 1870–1902. Zbl0963.60079MR1742892
  31. [31] S. Sagitov. The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 (1999) 1116–1125. Zbl0962.92026MR1742154
  32. [32] J. Schweinsberg. A necessary and sufficient condition for the Λ-coalescent to come down from infinity. Electron. Comm. Probab. 5 (2000) 1–11. Zbl0953.60072MR1736720
  33. [33] J. Schweinsberg. Coalescent processes obtained from supercritical Galton–Watson processes. Stochastic Process. Appl. 106 (2003) 107–139. Zbl1075.60571MR1983046
  34. [34] M. L. Silverstein. A new approach to local times. J. Math. Mech. 17 (1968) 1023–1054. Zbl0184.41101MR226734
  35. [35] R. Slack. A branching process with mean one and possibly infinite variance. Z. Wahrsch. Verw. Gebiete 9 (1968) 139–145. Zbl0164.47002MR228077
  36. [36] A. M. Yaglom. Certain limit theorems of the theory of branching processes. Dokl. Acad. Nauk SSSR 56 (1947) 795–798. Zbl0041.45602MR22045

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.