Small-time behavior of beta coalescents

Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason

doi:10.1214/07-AIHP103

Berestycki, Julien ; Berestycki, Nathanaël ; Schweinsberg, Jason

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 2, pp. 214-238.

Résumé
Abstract

L’objet de ce travail est l’étude du comportement asymptotique en temps petit des Beta-coalescents. Ces processus décrivent la limite d’échelle de la généalogie d’un certain nombre de modèles en génétique des populations. Nous donnons en particulier un théorème de convergence presque sûre pour le nombre de blocs renormalisé. Nous décrivons également le comportement asymptotique des tailles des blocs. Ces résultats permettent de calculer la dimension de Hausdorff et la dimension de packing d’un espace métrique associé à ce type de coalescents, ainsi que la longueur totale des branches de l’arbre de coalescence. Ce dernier résultat correspond à une question qui se pose en génétique des populations. Enfin, ces résultats sont en partie étendus par des arguments de couplage aux cas de $Λ$ -coalescents pour lesquels la mesure $Λ$ a un comportement près de 0 semblable à celui d’une distribution Beta. Les méthodes employées reposent essentiellement sur un lien entre Beta-coalescent et les processus de branchement à espace d’état continu.

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 $\int_{0}^{1} x^{k - 2 (1 - x)} b - k Λ (d x)$ . 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.

MR Zbl | 2 citations dans Numdam

DOI : 10.1214/07-AIHP103

Classification : 60J25, 60J85, 60J75, 60K99
Mots-clés : coalescence, continuous-state branching process, coalescent with multiple mergers

@article{AIHPB_2008__44_2_214_0,
     author = {Berestycki, Julien and Berestycki, Nathana\"el and Schweinsberg, Jason},
     title = {Small-time behavior of beta coalescents},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {214--238},
     publisher = {Gauthier-Villars},
     volume = {44},
     number = {2},
     year = {2008},
     doi = {10.1214/07-AIHP103},
     mrnumber = {2446321},
     zbl = {1214.60034},
     language = {en},
     url = {https://www.numdam.org/articles/10.1214/07-AIHP103/}
}

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
SP  - 214
EP  - 238
VL  - 44
IS  - 2
PB  - Gauthier-Villars
UR  - https://www.numdam.org/articles/10.1214/07-AIHP103/
DO  - 10.1214/07-AIHP103
LA  - en
ID  - AIHPB_2008__44_2_214_0
ER  -

%0 Journal Article
%A Berestycki, Julien
%A Berestycki, Nathanaël
%A Schweinsberg, Jason
%T Small-time behavior of beta coalescents
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2008
%P 214-238
%V 44
%N 2
%I Gauthier-Villars
%U https://www.numdam.org/articles/10.1214/07-AIHP103/
%R 10.1214/07-AIHP103
%G en
%F AIHPB_2008__44_2_214_0

Berestycki, Julien; Berestycki, Nathanaël; Schweinsberg, Jason. Small-time behavior of beta coalescents. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 2, pp. 214-238. doi : 10.1214/07-AIHP103. https://www.numdam.org/articles/10.1214/07-AIHP103/

Bibliographie
Cité par

[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. | MR | Zbl

[2] A.-L. Basdevant. Ruelle's probability cascades seen as a fragmentation process. Markov Process. Related Fields 12 (2006) 447-474. | MR | Zbl

[3] J. Berestycki, N. Berestycki and J. Schweinsberg. Beta-coalescents and continuous stable random trees. Ann. Probab. 35 (2007) 1835-1887. | MR | Zbl

[4] J. Bertoin. Lévy Processes. Cambridge University Press, Cambridge, 1996. | MR | Zbl

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

[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. | MR | Zbl

[7] J. Bertoin and J.-F. Le Gall. Stochastic flows associated to coalescent processes. Probab. Theory Related Fields 126 (2003) 261-288. | MR | Zbl

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

[9] J. Bertoin and J. Pitman. Two coalescents derived from the ranges of stable subordinators. Electron. J. Probab. 5 (2000) 1-17. | MR | Zbl

[10] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge University Press, Cambridge, 1987. | MR | Zbl

[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. | MR | Zbl

[12] E. Bolthausen and A.-S. Sznitman. On Ruelle's probability cascades and an abstract cavity method. Comm. Math. Phys. 197 (1998) 247-276. | MR | Zbl

[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. | MR | Zbl

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

[15] R. M. Dudley. Real Analysis and Probability. Wadsworth and Brooks/Cole, Pacific Grove, CA, 1989. | MR | Zbl

[16] T. Duquesne. A limit theorem for the contour process of conditioned Galton-Watson trees. Ann. Probab. 31 (2003) 996-1027. | MR | Zbl

[17] T. Duquesne and J.-F. Le Gall. Probabilistic and fractal aspects of Lévy trees. Probab. Theory Related Fields 131 (2005) 553-603. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[21] K. Falconer. Fractal Geometry: Mathematical Foundations and Applications, 2nd edition. Wiley, Hoboken, NJ, 2003. | MR | Zbl

[22] C. Goldschmidt and J. Martin. Random recursive trees and the Bolthausen-Sznitman coalescent. Electron. J. Probab. 10 (2005) 718-745. | MR | Zbl

[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. | MR | Zbl

[24] J. F. C. Kingman. The representation of partition structures. J. London Math. Soc. 18 (1978) 374-380. | MR | Zbl

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

[26] J. Lamperti. The limit of a sequence of branching processes. Z. Wahrsch. Verw. Gebiete 7 (1967) 271-288. | MR | Zbl

[27] J. Lamperti. Continuous state branching processes. Bull. Amer. Math. Soc. 73 (1967) 382-386. | MR | Zbl

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

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

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

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

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

[33] J. Schweinsberg. Coalescent processes obtained from supercritical Galton-Watson processes. Stochastic Process. Appl. 106 (2003) 107-139. | MR | Zbl

[34] M. L. Silverstein. A new approach to local times. J. Math. Mech. 17 (1968) 1023-1054. | MR | Zbl

[35] R. Slack. A branching process with mean one and possibly infinite variance. Z. Wahrsch. Verw. Gebiete 9 (1968) 139-145. | MR | Zbl

[36] A. M. Yaglom. Certain limit theorems of the theory of branching processes. Dokl. Acad. Nauk SSSR 56 (1947) 795-798. | MR | Zbl

Cité par Sources :