# The coalescent structure of continuous-time Galton-Watson trees

Simon Harris, Samuel Johnston, Matthew Roberts

Research output: Contribution to journalArticle

### Abstract

Take a continuous-time Galton-Watson tree. If the system survives until a large time $T$, then choose $k$ particles uniformly from those alive. What does the ancestral tree drawn out by these $k$ particles look like? Some special cases are known but we give a more complete answer. We concentrate on near-critical cases where the mean number of offspring is $1+\mu/T$ for some $\mu\in\R$, and show that a scaling limit exists as $T\to\infty$. Viewed backwards in time, the resulting coalescent process is topologically equivalent to Kingman's coalescent, but with an interesting and highly non-trivial time change depending roughly on how the size of the population fluctuates over time. We give explicit formulas for the distribution of the coalescent times, as well as a construction as a mixture of independent and identically distributed random variables. In general subcritical and supercritical cases it is not possible to give such explicit formulas, but we highlight the special case of birth-death processes.
Original language English 44 Annals of Applied Probability Accepted/In press - 4 Sep 2019

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Galton-Watson Tree
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### Cite this

The coalescent structure of continuous-time Galton-Watson trees. / Harris, Simon; Johnston, Samuel; Roberts, Matthew.

In: Annals of Applied Probability, 04.09.2019.

Research output: Contribution to journalArticle

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AB - Take a continuous-time Galton-Watson tree. If the system survives until a large time $T$, then choose $k$ particles uniformly from those alive. What does the ancestral tree drawn out by these $k$ particles look like? Some special cases are known but we give a more complete answer. We concentrate on near-critical cases where the mean number of offspring is $1+\mu/T$ for some $\mu\in\R$, and show that a scaling limit exists as $T\to\infty$. Viewed backwards in time, the resulting coalescent process is topologically equivalent to Kingman's coalescent, but with an interesting and highly non-trivial time change depending roughly on how the size of the population fluctuates over time. We give explicit formulas for the distribution of the coalescent times, as well as a construction as a mixture of independent and identically distributed random variables. In general subcritical and supercritical cases it is not possible to give such explicit formulas, but we highlight the special case of birth-death processes.

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