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An introduction to Galton-Watson trees and their local limits

Abstract : The main object of this course given in Hammamet (December 2014) is the so-called Galton-Watson process. We introduce in the first chapter of this course the framework of discrete random trees. We then use this framework to construct GW trees that describe the genealogy of a GW process. It is very easy to recover the GW process from the GW tree as it is just the number of individuals at each generation. We then give alternative proofs of classical results on GW processes using the tree formalism. We focus in particular on the extinction probability (which was the first question of F. Galton) and on the description of the processes conditioned on extinction or non extinction. In a second chapter, we focus on local limits of conditioned GW trees. In the critical and sub-critical cases, the population becomes a.s. extinct and the associated genealogical tree is finite. However, it has a small but positive probability of being large (this notion must be precised). The question that arises is to describe the law of the tree conditioned of being large, and to say what exceptional event has occurred so that the tree is not typical. A first answer to this question is due to H. Kesten who conditioned a GW tree to reach height n and look at the limit in distribution when n tends to infinity. There are however other ways of conditioning a tree to be large: conditioning on having many vertices, or many leaves... We present here very recent general results concerning this kind of problems due to the authors of this course and completed by results of X. He.
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https://hal.archives-ouvertes.fr/hal-01164661
Contributor : Romain Abraham <>
Submitted on : Friday, September 18, 2020 - 2:38:30 PM
Last modification on : Friday, January 15, 2021 - 5:31:36 PM

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  • HAL Id : hal-01164661, version 2
  • ARXIV : 1506.05571

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Romain Abraham, Jean-François Delmas. An introduction to Galton-Watson trees and their local limits. 2015. ⟨hal-01164661v2⟩

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