Convergence to the coalescent and its relation to the time back to the most recent common ancestor
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 m...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2008-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
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| Online Access: | https://dmtcs.episciences.org/3573/pdf |
| _version_ | 1797270453437333504 |
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| author | Martin Möhle |
| author_facet | Martin Möhle |
| author_sort | Martin Möhle |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-25T02:04:31Z |
| format | Article |
| id | doaj.art-dd2fea00560f4a3eb78c7a9b95826eb8 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:04:31Z |
| publishDate | 2008-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-dd2fea00560f4a3eb78c7a9b95826eb82024-03-07T14:36:56ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502008-01-01DMTCS Proceedings vol. AI,...Proceedings10.46298/dmtcs.35733573Convergence to the coalescent and its relation to the time back to the most recent common ancestorMartin Möhle0https://orcid.org/0000-0003-3078-4777Mathematisches Institut [Dusseldorf]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.https://dmtcs.episciences.org/3573/pdfsimultaneous multiple collisionsmost recent common ancestorabsorption timeancestral processcoalescentexchangeability[info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-ds] mathematics [math]/dynamical systems [math.ds][math.math-co] mathematics [math]/combinatorics [math.co] |
| spellingShingle | Martin Möhle Convergence to the coalescent and its relation to the time back to the most recent common ancestor Discrete Mathematics & Theoretical Computer Science simultaneous multiple collisions most recent common ancestor absorption time ancestral process coalescent exchangeability [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-ds] mathematics [math]/dynamical systems [math.ds] [math.math-co] mathematics [math]/combinatorics [math.co] |
| title | Convergence to the coalescent and its relation to the time back to the most recent common ancestor |
| title_full | Convergence to the coalescent and its relation to the time back to the most recent common ancestor |
| title_fullStr | Convergence to the coalescent and its relation to the time back to the most recent common ancestor |
| title_full_unstemmed | Convergence to the coalescent and its relation to the time back to the most recent common ancestor |
| title_short | Convergence to the coalescent and its relation to the time back to the most recent common ancestor |
| title_sort | convergence to the coalescent and its relation to the time back to the most recent common ancestor |
| topic | simultaneous multiple collisions most recent common ancestor absorption time ancestral process coalescent exchangeability [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-ds] mathematics [math]/dynamical systems [math.ds] [math.math-co] mathematics [math]/combinatorics [math.co] |
| url | https://dmtcs.episciences.org/3573/pdf |
| work_keys_str_mv | AT martinmohle convergencetothecoalescentanditsrelationtothetimebacktothemostrecentcommonancestor |