Weak convergence of the scaled jump chain and number of mutations of the Kingman coalescent
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2020
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| _version_ | 1866910874324697088 |
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| author | Favero, Martina Hult, Henrik |
| author_facet | Favero, Martina Hult, Henrik |
| contents | The Kingman coalescent is a fundamental process in population genetics modelling the ancestry of a sample of individuals backwards in time. In this paper, in a large-sample-size regime, we study asymptotic properties of the coalescent under neutrality and a general finite-alleles mutation scheme, i.e. including both parent independent and parent dependent mutation. In particular, we consider a sequence of Markov chains that is related to the coalescent and consists of block-counting and mutation-counting components. We show that these components, suitably scaled, converge weakly to deterministic components and Poisson processes with varying intensities, respectively. Along the way, we develop a novel approach, based on a change of measure, to generalise the convergence result from the parent independent to the parent dependent mutation setting, in which several crucial quantities are not known explicitly. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2011_06908 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Weak convergence of the scaled jump chain and number of mutations of the Kingman coalescent Favero, Martina Hult, Henrik Probability 60J90 (Primary) 60F05, 92D15 (Secondary) The Kingman coalescent is a fundamental process in population genetics modelling the ancestry of a sample of individuals backwards in time. In this paper, in a large-sample-size regime, we study asymptotic properties of the coalescent under neutrality and a general finite-alleles mutation scheme, i.e. including both parent independent and parent dependent mutation. In particular, we consider a sequence of Markov chains that is related to the coalescent and consists of block-counting and mutation-counting components. We show that these components, suitably scaled, converge weakly to deterministic components and Poisson processes with varying intensities, respectively. Along the way, we develop a novel approach, based on a change of measure, to generalise the convergence result from the parent independent to the parent dependent mutation setting, in which several crucial quantities are not known explicitly. |
| title | Weak convergence of the scaled jump chain and number of mutations of the Kingman coalescent |
| topic | Probability 60J90 (Primary) 60F05, 92D15 (Secondary) |
| url | https://arxiv.org/abs/2011.06908 |