Abstract
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 instanceb= 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.
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