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

作者：Limic Vlada^*; Talarczyk Anna

来源：Electronic Journal of Probability, 2015, 20(0): 1-20.

DOI：10.1214/EJP.v20-3818

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

We consider standard Lambda-coalescents (or coalescents with multiple collisions) with a non-trivial "Kingman part". Equivalently, 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 -> 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.

• 出版日期2015-4-18