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The time to absorption in Λ-coalescents

  • Götz Kersting (a1) and Anton Wakolbinger (a1)

Abstract

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.

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Copyright

Corresponding author

Institute of Mathematics, Goethe University Frankfurt, Robert Mayer Strasse 10, 60325 Frankfurt am Main, Germany. Email address: kersting@math.uni-frankfurt.de
Institute of Mathematics, Goethe University Frankfurt, Robert Mayer Strasse 10, 60325 Frankfurt am Main, Germany. Email address: wakolbinger@math.uni-frankfurt.de

References

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[1]Gnedin, A.,Iksanov, A., and Marynych, A. (2011).On Λ-coalescents with dust component.J. Appl. Prob. 48,11331151.
[2]Goldschmidt, C. and Martin, J. B. (2005).Random recursive trees and the Bolthausen‒Sznitman coalesent.Electron. J. Prob. 10,718745.
[3]Herriger, P. and Möhle, M. (2012).Conditions for excheangable coalescents to come down from infinity.ALEA 9,637665.
[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. 54,15271555.
[5]Möhle, M. (2014).On hitting probabilities of beta coalescents and absorption times of coalescents that come down from infinity.ALEA 11,141159.
[6]Pitman, J. (1999).Coalescents with multiple collisions.Ann. Prob. 27,18701902.
[7]Sagitov, S. (1999).The general coalescent with asynchronous mergers of ancestral lines.J. Appl. Prob. 36,11161125.
[8]Schweinsberg, J. (2000).A necessary and sufficient condition for the Λ-coalescent to come down from infinity.Electron. Commun. Prob. 5,111.

Keywords

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The time to absorption in Λ-coalescents

  • Götz Kersting (a1) and Anton Wakolbinger (a1)

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