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2015 Diffusion limits at small times for $\Lambda$-coalescents with a Kingman component
Vlada Limic, Anna Talarczyk
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Electron. J. Probab. 20: 1-20 (2015). DOI: 10.1214/EJP.v20-3818


We consider standard $\Lambda$-coalescents (or coalescents with multiple collisions) with a non-trivial "Kingman part". That is, the driving measure $\Lambda$ has an atom at $0; \Lambda(\{0\})= c > 0$. It is known that all such coalescents come down from infinity. Moreover, the number of blocks $N_t$ is asymptotic to $v(t) = 2/(ct)$ as $t\to 0$. In the present paper we investigate the second-order asymptotics of $N_t$ in the functional sense at small times. This complements our earlier results on the fluctuations of the number of blocks for a class of regular $\Lambda$-coalescents without the Kingman part. In the present setting it turns out that the Kingman part dominates and the limit process is a Gaussian diffusion, as opposed to the stable limit in our previous work.


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Vlada Limic. Anna Talarczyk. "Diffusion limits at small times for $\Lambda$-coalescents with a Kingman component." Electron. J. Probab. 20 1 - 20, 2015.


Accepted: 18 April 2015; Published: 2015
First available in Project Euclid: 4 June 2016

zbMATH: 1328.60094
MathSciNet: MR3339865
Digital Object Identifier: 10.1214/EJP.v20-3818

Primary: 60J25
Secondary: 60F17, 60G55, 60J60, 92D25


Vol.20 • 2015
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