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On asymptotics of the beta coalescents. (English) Zbl 1323.60021

Summary: We show that the total number of collisions in the exchangeable coalescent process driven by the \(\mathrm{beta}(1, b)\) measure converges in distribution to a 1-stable law, as the initial number of particles goes to \(\infty\). 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 \(\mathrm{beta}(a, b)\)-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 \(\mathrm{beta}(1, b)\)-coalescent by exploiting the method of sequential approximations.

MSC:

60C05 Combinatorial probability
60G09 Exchangeability for stochastic processes
60F05 Central limit and other weak theorems
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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