Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-21T00:09:36.771Z Has data issue: false hasContentIssue false
  • English
  • Français

Local convergence of critical Galton–Watson trees

Part of: Markov processes Probability theory on algebraic and topological structures Graph theory

Published online by Cambridge University Press:  30 November 2023

Aymen Bouaziz
Aymen Bouaziz*
Affiliation:
Université de Tunis El Manar
*
*Postal address: Aymen Bouaziz, Institut préparatoire aux études scientifiques et techniques, 2070 La Marsa, Tunisie. Email: bouazizaymen18@yahoo.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the local convergence of critical Galton–Watson trees under various conditionings. We give a sufficient condition, which serves to cover all previous known results, for the convergence in distribution of a conditioned Galton–Watson tree to Kesten’s tree. We also propose a new proof to give the limit in distribution of a critical Galton–Watson tree, with finite support, conditioned on having a large width.

Keywords

Galton–Watson treerandom treeslocal limitswidth

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60B10: Convergence of probability measures 05C05: Trees
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 7
Check for updates
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abraham, R., Bouaziz, A., and Delmas, J. F. (2017). Local limits of Galton–Watson trees conditioned of the number of the protected nodes. J. Appl. Prob. 54, 55–65.CrossRefGoogle Scholar
Abraham, R. and Delmas, J. F. (2014). Local limits of conditioned Galton–Watson trees: The infinite spine case. Electron. J. Prob. 19, 1–19.Google Scholar
Abraham, R. and Delmas, J. F. (2015). An introduction to Galton–Watson trees and their local limits. Preprint, arXiv:1506.05571.Google Scholar
He, X. (2017). Conditioning Galton–Watson trees on large maximal out-degree. J. Theoret. Prob. 30, 842–851.CrossRefGoogle Scholar
He, X. (2022). Local convergence of critical random trees and continuous-state branching processes. J. Theoret. Prob. 35, 685–713.CrossRefGoogle Scholar
Kesten, H. (1986). Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Prob. Statist. 22, 425–487.Google Scholar
Neveu, J. (1986). Arbres et processus de Galton–Watson. Ann. Inst. H. Poincaré Prob. Statist. 22, 199–207.Google Scholar