On the Coalescence Time Distribution in Multi-type Supercritical Branching Processes
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914387790397440 |
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| author | Krasnowska, Janique Jenkins, Paul Johansen, Adam |
| author_facet | Krasnowska, Janique Jenkins, Paul Johansen, Adam |
| contents | Consider a population evolving as a discrete-time supercritical multi-type Galton--Watson process. Suppose we run the process for $T$ generations, then sample $k$ individuals uniformly at generation $T$ and trace their genealogy backwards in time. In the limiting regime as $T \rightarrow \infty$, the expected behaviour of the sample's ancestry has been analysed extensively in the single-type case and, more recently, for multi-type processes in the critical case. In this paper, we present a formula for the distribution function of the generation $t$ of the most recent common ancestor in terms of the limiting distribution of the normalised population size. In addition, we provide effective bounds for the decay of this distribution function to 1 in terms of the harmonic moments of the population size at generation $t$. In order to better understand the behaviour of these harmonic moments, we use a multi-type generalisation of the Harris--Sevastyanov transformation to express harmonic moments at generation $t$ in terms of moments of the transformed process at the first generation. We present numerical results demonstrating that it is possible to approximate the coalescence time distribution effectively in practical settings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11990 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Coalescence Time Distribution in Multi-type Supercritical Branching Processes Krasnowska, Janique Jenkins, Paul Johansen, Adam Probability 60J85 (Primary) 60J90, 60J95 (Secondary) Consider a population evolving as a discrete-time supercritical multi-type Galton--Watson process. Suppose we run the process for $T$ generations, then sample $k$ individuals uniformly at generation $T$ and trace their genealogy backwards in time. In the limiting regime as $T \rightarrow \infty$, the expected behaviour of the sample's ancestry has been analysed extensively in the single-type case and, more recently, for multi-type processes in the critical case. In this paper, we present a formula for the distribution function of the generation $t$ of the most recent common ancestor in terms of the limiting distribution of the normalised population size. In addition, we provide effective bounds for the decay of this distribution function to 1 in terms of the harmonic moments of the population size at generation $t$. In order to better understand the behaviour of these harmonic moments, we use a multi-type generalisation of the Harris--Sevastyanov transformation to express harmonic moments at generation $t$ in terms of moments of the transformed process at the first generation. We present numerical results demonstrating that it is possible to approximate the coalescence time distribution effectively in practical settings. |
| title | On the Coalescence Time Distribution in Multi-type Supercritical Branching Processes |
| topic | Probability 60J85 (Primary) 60J90, 60J95 (Secondary) |
| url | https://arxiv.org/abs/2603.11990 |