Heavy Tails, Long-Range Dependence, and Beyond / Queues lourdes, dépendance de long terme et  au-delà

Collection Heavy Tails, Long-Range Dependence, and Beyond / Queues lourdes, dépendance de long terme et  au-delà

Organizer(s) Biermé, Hermine ; Kulik, Rafal ; Mikosch, Thomas ; Wang, Yizao ; Wintenberger, Olivier
Date(s) 7/4/22 - 7/8/22
linked URL https://conferences.cirm-math.fr/2633.html
00:00:00 / 00:00:00
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Power-laws and weak convergence of the Kingman coalescent

By Henrik Hult

The Kingman coalescent is a fundamental process in population genetics modelling the ancestry of a sample of individuals backwards in time. In this paper, weak convergence is proved for a sequence of Markov chains consisting of two components related to the Kingman coalescent, under a parent dependent d-alleles mutation scheme, as the sample size, grows to infinity. The first component is the normalised d-dimensional jump chain of the block counting processes of the Kingman coalescent. The second component is a d^2-dimensional process counting the number of mutations between types occurring in the Kingman coalescent. Time is scaled by the sample size. The limiting process consists of a deterministic d-dimensional component, describing the limit of the block counting jump chain, and d^2 independent Poisson processes with state-dependent intensities, exploding at the origin, describing the limit of the number of mutations. The weak convergence result is first proved, using a generator approach, in the setting of parent independent mutations. A change of measure argument is used to extend the weak convergence result to include parent dependent mutations.

Information about the video

Citation data

  • DOI 10.24350/CIRM.V.19937903
  • Cite this video Hult Henrik (7/4/22). Power-laws and weak convergence of the Kingman coalescent. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19937903
  • URL https://dx.doi.org/10.24350/CIRM.V.19937903



  • FAVERO, Martina et HULT, Henrik. Asymptotic behaviour of sampling and transition probabilities in coalescent models under selection and parent dependent mutations. Electronic Communications in Probability, 2022, vol. 27, p. 1-13. - http://dx.doi.org/10.1214/22-ECP472
  • FAVERO, Martina et HULT, Henrik. Weak convergence of the scaled jump chain and number of mutations of the Kingman coalescent. arXiv preprint arXiv:2011.06908, 2020. - https://doi.org/10.48550/arXiv.2011.06908

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