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

We consider standard Λ-coalescents (or coalescents with multiple collisions) with a non-trivial “Kingman part”. Equivalently, the driving measure Λ has an atom at 0; Λ({0}) = c > 0. It is known that all such coalescents come down from infinity. Moreover, the number of blocks Nt is asymptotic to v(t) = 2/(ct) as t → 0. In the present paper we investigate the second-order asymptotics of Nt 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 Λ-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.