On the size of the block of 1 for Ξ-coalescents with dust

Freund, Fabian; Möhle, Martin

doi:10.15559/17-VMSTA92

Modern Stochastics: Theory and Applications

On the size of the block of 1 for Ξ-coalescents with dust

Volume 4, Issue 4 (2017), pp. 407–425

Fabian Freund Martin Möhle

https://doi.org/10.15559/17-VMSTA92

Pub. online: 27 December 2017 Type: Research Article

Open Access

Received
28 August 2017

Revised
4 December 2017

Accepted
6 December 2017

Published
27 December 2017

Abstract

We study the frequency process $f_{1}$ of the block of 1 for a Ξ-coalescent Π with dust. If Π stays infinite, $f_{1}$ is a jump-hold process which can be expressed as a sum of broken parts from a stick-breaking procedure with uncorrelated, but in general non-independent, stick lengths with common mean. For Dirac-Λ-coalescents with $\varLambda =\delta _{p}$, $p\in [\frac{1}{2},1)$, $f_{1}$ is not Markovian, whereas its jump chain is Markovian. For simple Λ-coalescents the distribution of $f_{1}$ at its first jump, the asymptotic frequency of the minimal clade of 1, is expressed via conditionally independent shifted geometric distributions.

References

[1]

Abraham, R., Delmas, J.-F.: A construction of a β-coalescent via the pruning of binary trees. J. Appl. Probab. 50(3), 772–790 (2013)

[2]

Abraham, R., Delmas, J.-F.: β-coalescents and stable Galton-Watson trees. Alea 12(1), 451–476 (2015)

[3]

Aldous, D., Ibragimov, I., Jacod, J., Aldous, D.: Exchangeability and related topics. in Ecole d’Eté de Probabilités de Saint-Flour xiii 1983, Volume 1117 of Lecture Notes in Mathematics. Springer Berlin/Heidelberg 10, 1–198 (1985)

[4]

Amann, H., Escher, J., Brookfield, G.: Analysis vol. 3. Springer (2005)

[5]

Berestycki, J., Berestycki, N., Schweinsberg, J.: Small-time behavior of beta coalescents. Ann. Inst. H. Poincaré Probab. Statist. 44(2), 214–238 (2008)

[6]

Blum, M.G.B., François, O.: Minimal clade size and external branch length under the neutral coalescent. Adv. Appl. Probab. 37(3), 647–662 (2005)

[7]

Caliebe, A., Neininger, R., Krawczak, M., Rösler, U.: On the length distribution of external branches in coalescence trees: genetic diversity within species. Theor. Popul. Biol. 72(2), 245–252 (2007)

[8]

Dhersin, J.-S., Freund, F., Siri-Jégousse, A., Yuan, L.: On the length of an external branch in the beta-coalescent. Stoch. Process. Appl. 123(5), 1691–1715 (2013)

[9]

Eldon, B., Wakeley, J.: Coalescent processes when the distribution of offspring number among individuals is highly skewed. Genetics 172(4), 2621–2633 (2006)

[10]

Erdös, P.: On a family of symmetric Bernoulli convolutions. American Journal of Mathematics 61(4), 974–976 (1939)

[11]

Freund, F.: Almost sure asymptotics for the number of types for simple Xi-coalescents. Electron. Commun. Probab. 17, 1–11 (2012)

[12]

Freund, F., Möhle, M.: On the number of allelic types for samples taken from exchangeable coalescents with mutation. Adv. Appl. Probab. 41(4), 1082–1101 (2009)

[13]

Freund, F., Möhle, M.: On the time back to the most recent common ancestor and the external branch length of the Bolthausen-Sznitman coalescent. Markov Process. Related Fields 15(3), 387–416 (2009)

[14]

Freund, F., Siri-Jégousse, A.: Minimal clade size in the Bolthausen-Sznitman coalescent. J. Appl. Probab. 51(3), 657–668 (2014)

[15]

Gnedin, A., Iksanov, A., Marynych, A.: Λ-coalescents: a survey. J. Appl. Probab. 51A, 23–40 (2014)

[16]

Gnedin, A., Iksanov, A., Möhle, M.: On asymptotics of exchangeable coalescents with multiple collisions. J. Appl. Probab. 45(4), 1186–1195 (2008)

[17]

Goldschmidt, C., Martin, J.B.: Random recursive trees and the Bolthausen-Sznitman coalescent. Electron. J. Probab. 10(21), 718–745 (2005)

[18]

Grabner, P.J., Prodinger, H.: Asymptotic analysis of the moments of the Cantor distribution. Statist. Probab. Letters 26(3), 243–248 (1996)

[19]

Hénard, O.: The fixation line in the Λ-coalescent. Ann. Appl. Probab. 25(5), 3007–3032 (2015)

[20]

Herriger, P., Möhle, M.: Conditions for exchangeable coalescents to come down from infinity. Alea 9(2), 637–665 (2012)

[21]

Huillet, T., Möhle, M.: Asymptotics of symmetric compound Poisson population models. Combin. Probab. Comput. 24(1), 216–253 (2015)

[22]

Janson, S., Kersting, G.: On the total external length of the Kingman coalescent. Electron. J. Probab. 16, 2203–2218 (2011)

[23]

Kersting, G., Schweinsberg, J., Wakolbinger, A.: The size of the last merger and time reversal in Λ-coalescents. ArXiv e-prints (2017). 1701.00549

[24]

Kingman, J.F.C.: The coalescent. Stoch. Process. Appl. 13(3), 235–248 (1982)

[25]

Kingman, J.F.C.: Poisson Processes. Wiley (1993)

[26]

Lad, F., Taylor, W.F.C.: The moments of the Cantor distribution. Statist. Probab. Letters 13(4), 307–310 (1992)

[27]

Limic, V.: On the speed of coming down from infinity for Ξ-coalescent processes. Electron. J. Probab. 15, 217–240 (2010)

[28]

Möhle, M.: Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson-Dirichlet coalescent. Stoch. Process. Appl. 120(11), 2159–2173 (2010)

[29]

Möhle, M.: On hitting probabilities of beta coalescents and absorption times of coalescents that come down from infinity. Alea 11, 141–159 (2014)

[30]

Möhle, M., Sagitov, S.: A classification of coalescent processes for haploid exchangeable population models. Ann. Probab. 29(4), 1547–1562 (2001)

[31]

Peres, Y., Schlag, W., Solomyak, B.: Sixty years of Bernoulli convolutions. Progress in probability, 39–68 (2000)

[32]

Pitman, J.: Coalescents with multiple collisions. Ann. Probab. 27(4), 1870–1902 (1999)

[33]

Schweinsberg, J.: Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5, 1–50 (2000)

[34]

Siri-Jégousse, A., Yuan, L.: Asymptotics of the minimal clade size and related functionals of certain beta-coalescents. Acta Appl. Math. 142(1), 127–148 (2016)

[35]

Solomyak, B.: On the random series $\sum \pm {\lambda }^{n}$ (an Erdös problem). Annals of Mathematics, 611–625 (1995)

Full article Related articles

Open access article under the CC BY license.

Keywords

Ξ-coalescent coalescent with dust Poisson point process minimal clade exchangeability

MSC2010

60F15 60J75 60G55 60G09

Metrics

since February 2017

Article info
views

Full article
views

PDF
downloads

XML
downloads

RSS

Authors

Abstract

References

Export citation

Copy and paste formatted citation

Download citation in file