We show that the total number of collisions in the exchangeable coalescent process driven by the beta(1, b) measure converges in distribution to a 1-stable law, as the initial number of particles goes to ∞. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance b = 1, which corresponds to the Bolthausen-Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta(a, b)-coalescents with 0 < a < 1 leads to a simplified derivation of the known (2 - a)-stable limit. We furthermore derive asymptotic expansions for the moments of the number of collisions and of the total branch length for the beta(1, b)-coalescent by exploiting the method of sequential approximations.
"On asymptotics of the beta coalescents." Adv. in Appl. Probab. 46 (2) 496 - 515, June 2014. https://doi.org/10.1239/aap/1401369704