Convergence to the coalescent and its relation to the time back to the most recent common ancestor. (English) Zbl 1355.92076

Möhle, Martin

Fifth colloquium on mathematics and computer science. Lectures from the colloquium, Blaubeuren, Germany, September 22–26, 2008. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). Discrete Mathematics and Theoretical Computer Science Proceedings AI, 311-324 (2008).

Summary: For the class of haploid exchangeable population models with non-overlapping generations and population size \(N\) it is shown that, as \(N\) tends to infinity, convergence of the time-scaled 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.
For the entire collection see [Zbl 1172.05004].

MSC:

92D15

Problems related to evolution

Keywords:

absorption time; ancestral process; coalescent; exchangeability; most recent common ancestor; simultaneous multiple collisions