Second-order asymptotics for the block counting process in a class of regularly varying .-coalescents

Consider a standard .-coalescent that comes down from infinity.Such a coalescent starts from a configuration consisting of infinitely many blocks at time 0, but its number of blocks Nt is a finite random variable at each positive time t.Berestycki et al. [Ann. Probab.38 (2010) 207.233] found the first-order approximation v for the process N at small times.This is a deterministic function satisfying Nt/vt.1 as t.0.The present paper reports on the first progress in the study of the second-order asymptotics for N at small times.We show that, if the driving measure . has a density near zero which behaves as x.. with ..(0,1), then the process (..1/(1+.)(N.t/v.t.1))t.0 converges in law as ..0 in the Skorokhod space to a totally skewed (1+.)-stable process.Moreover, this process is a unique solution of a related stochastic differential equation of Ornstein.Uhlenbeck type, with a completely asymmetric stable Lévy noise