### Abstract

Original language | English |
---|---|

Number of pages | 44 |

Journal | Annals of Applied Probability |

Publication status | Accepted/In press - 4 Sep 2019 |

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### Cite this

*Annals of Applied Probability*.

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

Research output: Contribution to journal › Article

*Annals of Applied Probability*.

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TY - JOUR

T1 - The coalescent structure of continuous-time Galton-Watson trees

AU - Harris, Simon

AU - Johnston, Samuel

AU - Roberts, Matthew

PY - 2019/9/4

Y1 - 2019/9/4

N2 - 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.

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.

M3 - Article

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

ER -