Limit distributions and random trees derived from the birthday problem with unequal probabilities.
Camarri, Michael, Pitman, Jim (2000)
Electronic Journal of Probability [electronic only]
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Camarri, Michael, Pitman, Jim (2000)
Electronic Journal of Probability [electronic only]
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David Croydon (2008)
Annales de l'I.H.P. Probabilités et statistiques
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In this article it is shown that the brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete -vertex ordered graph trees whose search-depth functions converge to the brownian excursion as →∞. We prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of our main result. The assumptions of the article cover the important example of simple random...
Geiger, Jochen (2000)
Electronic Journal of Probability [electronic only]
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David Aldous, Jim Pitman (1998)
Annales de l'I.H.P. Probabilités et statistiques
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Grégory Miermont (2008)
Annales de l'I.H.P. Probabilités et statistiques
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We prove that critical multitype Galton–Watson trees converge after rescaling to the brownian continuum random tree, under the hypothesis that the offspring distribution is irreducible and has finite covariance matrices. Our study relies on an ancestral decomposition for marked multitype trees, and an induction on the number of types. We then couple the genealogical structure with a spatial motion, whose step distribution may depend on the structure of the tree in a local way, and show...
Jean-François Le Gall (2006)
Annales de la faculté des sciences de Toulouse Mathématiques
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We survey recent developments about random real trees, whose prototype is the Continuum Random Tree (CRT) introduced by Aldous in 1991. We briefly explain the formalism of real trees, which yields a neat presentation of the theory and in particular of the relations between discrete Galton-Watson trees and continuous random trees. We then discuss the particular class of self-similar random real trees called stable trees, which generalize the CRT. We review several important results concerning...