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

Abstract

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.