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Abstract: This article considers a model of genealogy corresponding to a regularexchangeable coalescent also known as Xi-coalescent started from a largefinite configuration, and undergoing neutral mutations. Asymptotic expressionsfor the number of active lineages were obtained by the author in a previouswork. Analogous results for the number of active mutation-free lineages and thecombined lineage lengths are derived using the same martingale-based technique.They are given in terms of convergence in probability, while extensions toconvergence in moments and convergence almost surely are discussed. The abovementioned results have direct consequences on the sampling theory in theXi-coalescent setting. In particular, the regular Xi-coalescents that come downfrom infinity i.e., with locally finite genealogies, have an asymptoticallyequal number of families under the corresponding infinite alleles and infinitesites models. In special cases, quantitative asymptotic formulae for the numberof families that contain a fixed number of individuals can be given.



Author: Vlada Limic

Source: https://arxiv.org/







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