The Annals of Applied Probability

A small-time coupling between $\Lambda$-coalescents and branching processes

Julien Berestycki, Nathanaël Berestycki, and Vlada Limic

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We describe a new general connection between $\Lambda$-coalescents and genealogies of continuous-state branching processes. This connection is based on the construction of an explicit coupling using a particle representation inspired by the lookdown process of Donnelly and Kurtz. This coupling has the property that the coalescent comes down from infinity if and only if the branching process becomes extinct, thereby answering a question of Bertoin and Le Gall. The coupling also offers new perspective on the speed of coming down from infinity and allows us to relate power-law behavior for $N^{\Lambda}(t)$ to the classical upper and lower indices arising in the study of pathwise properties of Lévy processes.

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Ann. Appl. Probab., Volume 24, Number 2 (2014), 449-475.

First available in Project Euclid: 10 March 2014

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Primary: 60J25: Continuous-time Markov processes on general state spaces 60F99: None of the above, but in this section 92D25: Population dynamics (general)

$\Lambda$-coalescents continuous-state branching processes particle system representation Lévy processes lookdown construction Fleming–Viot processes


Berestycki, Julien; Berestycki, Nathanaël; Limic, Vlada. A small-time coupling between $\Lambda$-coalescents and branching processes. Ann. Appl. Probab. 24 (2014), no. 2, 449--475. doi:10.1214/12-AAP911.

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