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

Consider a standard ${\Lambda }$-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 $N_t$ 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 $N_t/v_t\to1$ as $t\to0$. 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 $\Lambda$ has a density near zero which behaves as $x^{-\beta}$ with $\beta\in(0,1)$, then the process $(\varepsilon^{-1/(1+\beta)}(N_{\varepsilon t}/v_{\varepsilon t}-1))_{t\ge0}$ converges in law as $\varepsilon\to0$ in the Skorokhod space to a totally skewed $(1+\beta)$-stable process. Moreover, this process is a unique solution of a related stochastic differential equation of Ornstein-Uhlenbeck type, with a completely asymmetric stable L\'{e}vy noise.