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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2401.02233 |
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| _version_ | 1866910287382183936 |
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| author | Gui, Toni |
| author_facet | Gui, Toni |
| contents | We study a model of a population with individuals sampled from different species. The Yule-$Λ$ nested coalescent describes the genealogy of the sample when each species merges with another randomly chosen species with a constant rate $c$ and the mergers of individuals in each species follow the $Λ$-coalescent. For the Yule-$Λ$ nested coalescent with $c<\int_0^1x^{-1}Λ(dx)<\infty$, where $Λ$ is the measure that characterizes the $Λ$-coalescent, we show that under some initial conditions, the distribution of the number of individual lineages belonging to one species converges weakly to the distribution $ν_c^*$, which is the solution to some recursive distributional equation (RDE) with finite mean. In addition, we show that for some values of $c$, the RDE has another solution with infinite mean. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02233 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Yule-$Λ$ Nested Coalescent: Distribution of the Number of Lineages Gui, Toni Probability 60J90 60J95 We study a model of a population with individuals sampled from different species. The Yule-$Λ$ nested coalescent describes the genealogy of the sample when each species merges with another randomly chosen species with a constant rate $c$ and the mergers of individuals in each species follow the $Λ$-coalescent. For the Yule-$Λ$ nested coalescent with $c<\int_0^1x^{-1}Λ(dx)<\infty$, where $Λ$ is the measure that characterizes the $Λ$-coalescent, we show that under some initial conditions, the distribution of the number of individual lineages belonging to one species converges weakly to the distribution $ν_c^*$, which is the solution to some recursive distributional equation (RDE) with finite mean. In addition, we show that for some values of $c$, the RDE has another solution with infinite mean. |
| title | The Yule-$Λ$ Nested Coalescent: Distribution of the Number of Lineages |
| topic | Probability 60J90 60J95 |
| url | https://arxiv.org/abs/2401.02233 |