Electronic Journal of Probability

Diffusion limits at small times for $\Lambda$-coalescents with a Kingman component

Vlada Limic and Anna Talarczyk

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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.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 45, 20 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60F17: Functional limit theorems; invariance principles 92D25: Population dynamics (general) 60J60: Diffusion processes [See also 58J65] 60G55: Point processes

Kingman coalescent $\Lambda$-coalescent coming down from infinity functional limit theorems diffusion processes Poisson random measure

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Limic, Vlada; Talarczyk, Anna. Diffusion limits at small times for $\Lambda$-coalescents with a Kingman component. Electron. J. Probab. 20 (2015), paper no. 45, 20 pp. doi:10.1214/EJP.v20-3818. https://projecteuclid.org/euclid.ejp/1465067151

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