Discrete Mathematics & Theoretical Computer Science 
For the class of haploid exchangeable population models with nonoverlapping generations and population size $N$ it is shown that, as $N$ tends to infinity, convergence of the timescaled ancestral process to Kingman's coalescent and convergence in distribution of the scaled times back to the most recent common ancestor (MRCA) to the corresponding times back to the MRCA of the Kingman coalescent are equivalent. Extensions of this equivalence are derived for exchangeable population models being in the domain of attraction of a coalescent process with multiple collisions. The proofs are based on the property that the total rates of a coalescent with multiple collisions already determine the distribution of the coalescent. It is finally shown that similar results cannot be obtained for the full class of exchangeable coalescents allowing for simultaneous multiple collisions of ancestral lineages, essentially because the total rates do not determine the distribution of a general exchangeable coalescent.
Source : ScholeXplorer
IsRelatedTo DOI 10.1007/s0028500302186 Source : ScholeXplorer IsRelatedTo PMID 14523577
