%0 Generic %1 gnedin2012asymptotics %A Gnedin, Alexander %A Iksanov, Alexander %A Marynych, Alexander %A Moehle, Martin %D 2012 %K beta_coalescent coalescent_theory lambda_coalescent large-sample_properties %T On asymptotics of the beta-coalescents %U http://arxiv.org/abs/1203.3110 %X 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 infinity. 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.

@misc{gnedin2012asymptotics, 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 infinity. 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.}, added-at = {2013-12-14T06:46:12.000+0100}, author = {Gnedin, Alexander and Iksanov, Alexander and Marynych, Alexander and Moehle, Martin}, biburl = {https://www.bibsonomy.org/bibtex/284eebc2a1baa2a3902c681c7cbaca0ae/peter.ralph}, description = {[1203.3110] On asymptotics of the beta-coalescents}, interhash = {116fb4031e469f0cf625056afeb1d530}, intrahash = {84eebc2a1baa2a3902c681c7cbaca0ae}, keywords = {beta_coalescent coalescent_theory lambda_coalescent large-sample_properties}, note = {cite arxiv:1203.3110}, timestamp = {2013-12-14T06:46:12.000+0100}, title = {On asymptotics of the beta-coalescents}, url = {http://arxiv.org/abs/1203.3110}, year = 2012 }