Journal ArticleDOI
On the profile of random trees
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TLDR
In this paper, a plane rooted tree with n nodes is regarded as family tree of a Galton-Watson branching process conditioned on the total progeny, and it is shown that these two processes converge weakly to Brownian excursion local time.Abstract:
Let T be a plane rooted tree with n nodes which is regarded as family tree of a Galton-Watson branching process conditioned on the total progeny. The profile of the tree may be described by the number of nodes or the number of leaves in layer , respectively. It is shown that these two processes converge weakly to Brownian excursion local time. This is done via characteristic functions obtained by means of generating functions arising from the combinatorial setup and complex contour integration. Besides, an integral representation for the two-dimensional density of Brownian excursion local time is derived. © 1997 John Wiley & Sons, Inc. Random Struct. Alg.,10, 421–451, 1997read more
Citations
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Journal ArticleDOI
Convergence of Probability Measures
TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Journal ArticleDOI
Random trees and applications
TL;DR: An introduction to the theory of the Brownian snake, which combines the genealogical structure of random real trees with independent spatial motions, and some applications to a class of semilinear partial differential equations.
Journal IssueDOI
Random cutting and records in deterministic and random trees
TL;DR: The number of cuts is equal (in distribution) to the number of records in the tree when edges (or vertices) are assigned random labels when edges are assignedrandom labels.
Journal ArticleDOI
The depth first processes of Galton--Watson trees converge to the same Brownian excursion
TL;DR: In this paper, a strong relation between the depth first processes associated to Galton-Watson trees with finite variance, conditioned by the total progeny, was shown, and it was shown that these processes (suitably normalized) converge to the same Brownian excursion.
Journal ArticleDOI
Random real trees
TL;DR: In this article, the authors introduce le formalisme d'arbre reel aleatoires, which fournit une presentation elegante de la theorie, and present plusieurs resultats importants au sujet des arbres stables, notamment leur propriete de branchement, analogue continu d'une propriete bien connue, and le calcul de leurs dimensions fractales.
References
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Book
Convergence of Probability Measures
TL;DR: Weak Convergence in Metric Spaces as discussed by the authors is one of the most common modes of convergence in metric spaces, and it can be seen as a form of weak convergence in metric space.
Book
Brownian Motion and Stochastic Calculus
TL;DR: In this paper, the authors present a characterization of continuous local martingales with respect to Brownian motion in terms of Markov properties, including the strong Markov property, and a generalized version of the Ito rule.
Book
Continuous martingales and Brownian motion
Daniel Revuz,Marc Yor +1 more
TL;DR: In this article, the authors present a comprehensive survey of the literature on limit theorems in distribution in function spaces, including Girsanov's Theorem, Bessel Processes, and Ray-Knight Theorem.
Journal ArticleDOI
Convergence of Probability Measures
TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Journal ArticleDOI
The Continuum Random Tree III
TL;DR: The notion of convergence in distribution was introduced in this paper, which is based on the assumption that, for fixed k, the subtrees of a random tree determined by k randomly chosen vertices converge to a limit continuum random tree.