Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent
We consider a coalescent process as a model for the genealogy of a sample from a population. The population is subject to neutral mutation at constant rate p per individual and every mutation gives rise to a completely new type. The allelic partition is obtained by tracing back to the most recent mu...
| Main Authors: | , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
2008
|
| _version_ | 1826280042285498368 |
|---|---|
| author | Basdevant, A Goldschmidt, C |
| author_facet | Basdevant, A Goldschmidt, C |
| author_sort | Basdevant, A |
| collection | OXFORD |
| description | We consider a coalescent process as a model for the genealogy of a sample from a population. The population is subject to neutral mutation at constant rate p per individual and every mutation gives rise to a completely new type. The allelic partition is obtained by tracing back to the most recent mutation for each individual and grouping together individuals whose most recent mutations are the same. The allele frequency spectrum is the sequence (N 1(n), N2(n),..., Nn(n)), where Nk(n) is number of blocks of size k in the allelic partition with sample size n. In this paper, we prove law of large numbers-type results for the allele frequency spectrum when the coalescent process is taken to be the Bolthausen-Sznitman coalescent. In particular, we show that n-1 (log n)N1(n) →p; ρ and, for k ≥ 2, n-1(log n) 2Nk(n) →p ρ/(k(k - 1)) as n → ∞. Our method of proof involves tracking the formation of the allelic partition using a certain Markov process, for which we prove a fluid limit. |
| first_indexed | 2024-03-07T00:07:47Z |
| format | Journal article |
| id | oxford-uuid:7825a654-4b05-44c4-acfe-6d12dd3f8a67 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T00:07:47Z |
| publishDate | 2008 |
| record_format | dspace |
| spelling | oxford-uuid:7825a654-4b05-44c4-acfe-6d12dd3f8a672022-03-26T20:28:44ZAsymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescentJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:7825a654-4b05-44c4-acfe-6d12dd3f8a67EnglishSymplectic Elements at Oxford2008Basdevant, AGoldschmidt, CWe consider a coalescent process as a model for the genealogy of a sample from a population. The population is subject to neutral mutation at constant rate p per individual and every mutation gives rise to a completely new type. The allelic partition is obtained by tracing back to the most recent mutation for each individual and grouping together individuals whose most recent mutations are the same. The allele frequency spectrum is the sequence (N 1(n), N2(n),..., Nn(n)), where Nk(n) is number of blocks of size k in the allelic partition with sample size n. In this paper, we prove law of large numbers-type results for the allele frequency spectrum when the coalescent process is taken to be the Bolthausen-Sznitman coalescent. In particular, we show that n-1 (log n)N1(n) →p; ρ and, for k ≥ 2, n-1(log n) 2Nk(n) →p ρ/(k(k - 1)) as n → ∞. Our method of proof involves tracking the formation of the allelic partition using a certain Markov process, for which we prove a fluid limit. |
| spellingShingle | Basdevant, A Goldschmidt, C Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent |
| title | Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent |
| title_full | Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent |
| title_fullStr | Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent |
| title_full_unstemmed | Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent |
| title_short | Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent |
| title_sort | asymptotics of the allele frequency spectrum associated with the bolthausen sznitman coalescent |
| work_keys_str_mv | AT basdevanta asymptoticsoftheallelefrequencyspectrumassociatedwiththebolthausensznitmancoalescent AT goldschmidtc asymptoticsoftheallelefrequencyspectrumassociatedwiththebolthausensznitmancoalescent |