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)

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Keywords

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

MSC2010

60F15 60J75 60G55 60G09

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