The depth first processes of Galton--Watson trees converge to the same Brownian excursion

By Jean-François Marckert, Abdelkader Mokkadem. 2003

112 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/acc13052262ec8ffb43795a2201d8ca912e85afe

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Citations in the corpus

Random real trees (2006, methodology): “A simple approach to the convergence of the rescaled contour functions towards the normalized Brownian excursion was provided in [51].”
Recent progress in coalescent theory (2009, unknown): “Further details can be found in Aldous [3], but see also Marckert and Mokkadem [117] and Le Gall and Le Jan [114].”
Some families of increasing planar maps (2007, unknown): “In a ternary tree, the process giving the successive height of the nodes of degree 3 coincides with H ◦. Using [ 38 ], one check easily that”
The continuum limit of critical random graphs (2009, methodology): “An essential tool in what follows will be the following estimate on the distance between the depth-first walk and the height process (Theorem 3 of [39]).”
Random Trees, Lévy Processes and Spatial Branching Processes (2005, background): “This is of course reminiscent of a result of Aldous [3] who proved that in the case of a critical offspring distribution μ with finite variance, the contour process of a μ-Galton-Watson tree conditioned to have exactly p vertices converges after a suitable rescaling towards a normalized Brownian exc”
A Simple Proof of Duquesne’s Theorem on Contour Processes of Conditioned Galton–Watson Trees (2011, background): “In [21], under an additional exponential moment condition, Marckert & Mokkadem showed that the rescaled Lukasiewicz path, height function and contour function all converge in distribution to the same Brownian excursion.”
Rescaled bipartite planar maps converge to the Brownian map (2013, methodology): “In the case of Galton-Watson trees with a fixed size, the fact that the height process and the contour function converge jointly to the same Brownian excursion is due to Marckert and Mokkadem in [13].”
Itô’s excursion theory and random trees (2010, result): “Our proof is close in spirit to the one provided by Marckert and Mokkadem [21] under the assumption that the offspring distribution has exponential moments (see also Section 1.5 in [19]).”
Random trees and applications (2005, background): “The derivation is quite simple if the conditioning is “non-degenerate”: For instance if the tree is conditioned to have height greater than n (resp. total progeny greater than n), the scaling limit of the height function will be a Brownian excursion conditioned to have height greater than 1 (resp.…”
Combinatorial Stochastic Processes (2006, background): “Marckert and Mokkadem [293] derive Theorem 6.”