Second-order asymptotics for the block counting process in a class of regularly varying ${\Lambda}$-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 ... 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 ..., then the process ... converges in law as ... in the Skorokhod space to a totally skewed ...-stable process. Moreover, this process is a unique solution of a related stochastic differential equation of Ornstein-Uhlenbeck type, with a completely asymmetric stable Levy noise. (ProQuest: ... denotes formulae/symbols omitted.)