The Moran model with random resampling rates
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909416268234752 |
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| author | Athreya, Siva Hollander, Frank den Röllin, Adrian |
| author_facet | Athreya, Siva Hollander, Frank den Röllin, Adrian |
| contents | In this paper we consider the two-type Moran model with $N$ individuals. Each individual is assigned a resampling rate, drawn independently from a probability distribution ${\mathbb P}$ on ${\mathbb R}_+$, and a type, either $1$ or $0$. Each individual resamples its type at its assigned rate, by adopting the type of an individual drawn uniformly at random. Let $Y^N(t)$ denote the empirical distribution of the resampling rates of the individuals with type $1$ at time $Nt$. We show that if ${\mathbb P}$ has countable support and satisfies certain tail and moment conditions, then in the limit as $N\to\infty$ the process $(Y^N(t))_{t \geq 0}$ converges in law to the process $(S(t)\,¶)_{t \geq 0}$, in the so-called Meyer-Zheng topology, where $(S(t))_{t \geq 0}$ is the Fisher-Wright diffusion with diffusion constant $D$ given by $1/D = \int_{{\mathbb R}_+} (1/r)\,{\mathbb P}(\mathrm{d} r)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_01333 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Moran model with random resampling rates Athreya, Siva Hollander, Frank den Röllin, Adrian Probability 60J70, 60K35, 92D25 In this paper we consider the two-type Moran model with $N$ individuals. Each individual is assigned a resampling rate, drawn independently from a probability distribution ${\mathbb P}$ on ${\mathbb R}_+$, and a type, either $1$ or $0$. Each individual resamples its type at its assigned rate, by adopting the type of an individual drawn uniformly at random. Let $Y^N(t)$ denote the empirical distribution of the resampling rates of the individuals with type $1$ at time $Nt$. We show that if ${\mathbb P}$ has countable support and satisfies certain tail and moment conditions, then in the limit as $N\to\infty$ the process $(Y^N(t))_{t \geq 0}$ converges in law to the process $(S(t)\,¶)_{t \geq 0}$, in the so-called Meyer-Zheng topology, where $(S(t))_{t \geq 0}$ is the Fisher-Wright diffusion with diffusion constant $D$ given by $1/D = \int_{{\mathbb R}_+} (1/r)\,{\mathbb P}(\mathrm{d} r)$. |
| title | The Moran model with random resampling rates |
| topic | Probability 60J70, 60K35, 92D25 |
| url | https://arxiv.org/abs/2402.01333 |