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The time to absorption in Λ-coalescents

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  01 February 2019

Götz Kersting  and
Anton Wakolbinger
Götz Kersting*
Affiliation:
Goethe University Frankfurt
Anton Wakolbinger*
Affiliation:
Goethe University Frankfurt
*
Institute of Mathematics, Goethe University Frankfurt, Robert Mayer Strasse 10, 60325 Frankfurt am Main, Germany. Email address: [email protected]
Institute of Mathematics, Goethe University Frankfurt, Robert Mayer Strasse 10, 60325 Frankfurt am Main, Germany. Email address: [email protected]
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Abstract

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We present a law of large numbers and a central limit theorem for the time to absorption of Λ-coalescents with dust started from n blocks, as n→∞. The proofs rely on an approximation of the logarithm of the block-counting process by means of a drifted subordinator.

Keywords

Coalescenttime to absorptionlaw of large numberscentral limit theoremsubordinator with drift

MSC classification

Primary: 60J75: Jump processes
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60F05: Central limit and other weak theorems
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue A: Branching and Applied Probability , December 2018 , pp. 177 - 190
Copyright
Copyright © Applied Probability Trust 2018 

References

[1]Gnedin, A.,Iksanov, A., and Marynych, A. (2011).On Λ-coalescents with dust component.J. Appl. Prob. 48,1133–1151.Google Scholar
[2]Goldschmidt, C. and Martin, J. B. (2005).Random recursive trees and the Bolthausen‒Sznitman coalesent.Electron. J. Prob. 10,718–745.Google Scholar
[3]Herriger, P. and Möhle, M. (2012).Conditions for excheangable coalescents to come down from infinity.ALEA 9,637–665.Google Scholar
[4]Kersting, G.,Schweinsberg, J., and Wakolbinger, A. (2018).The size of the last merger and time reversal in Λ-coalescents.Ann. Inst. H. Poincaré Prob. Statist. 54,1527–1555.Google Scholar
[5]Möhle, M. (2014).On hitting probabilities of beta coalescents and absorption times of coalescents that come down from infinity.ALEA 11,141–159.Google Scholar
[6]Pitman, J. (1999).Coalescents with multiple collisions.Ann. Prob. 27,1870–1902.Google Scholar
[7]Sagitov, S. (1999).The general coalescent with asynchronous mergers of ancestral lines.J. Appl. Prob. 36,1116–1125.Google Scholar
[8]Schweinsberg, J. (2000).A necessary and sufficient condition for the Λ-coalescent to come down from infinity.Electron. Commun. Prob. 5,1–11.Google Scholar