A limit theorem for the contour process of condidtioned Galton--Watson trees

By Thomas Duquesne. 2003

193 citations recorded by the source corpus.

In this work, we study asymptotics of the genealogy of Galton--Watson processes conditioned on the total progeny. We consider a fixed, aperiodic and critical offspring distribution such that the rescaled Galton--Watson processes converges to a continuous-state branching process (CSBP) with a stable branching mechanism of index α∈(1,2]. We code the genealogy by two different processes: the contour process and the height process that Le Gall and Le Jan recently introduced. We show that the rescaled height process of the corresponding Galton--Watson family tree, with one ancestor and conditioned on the total progeny, converges in a functional sense, to a new process: the normalized excursion of the continuous height process associated with the α-stable CSBP. We deduce from this convergence an analogous limit theorem for the contour process. In the Brownian case α=2, the limiting process is the normalized Brownian excursion that codes the continuum random tree: the result is due to Aldous who used a different method.

Subfields: Artificial Intelligence, Condensed Matter Physics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cc467ed8d64d8c118920977151d03d900c29ba33

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Citations in the corpus

A Simple Proof of Duquesne’s Theorem on Contour Processes of Conditioned Galton–Watson Trees (2011, unknown): “We follow the presentation of [7].”
Self-similar fragmentations derived from the stable tree I (2003, methodology): “This section is mainly inspired by [14, 13 ].”
The gap between Gromov-vague and Gromov–Hausdorff-vague topology (2014, methodology): “This approach has been successfully applied to branching forests with a particular offspring distribution (see, for example, [14,13,22]).”
Sub-exponential tail bounds for conditioned stable Bienaymé–Galton–Watson trees (2015, background): “Duquesne [18] (see also [30]) showed that the convergence ( 1 Bn · W nt (tn), n Bn · H nt (tn) )”
Scaling limits of Markov branching trees with applications to Galton–Watson and random unordered trees (2010, methodology): “While [17] relies on quite refined aspects of Galton-Watson trees and their encodings by stochastic processes, our approach requires only to have some kind of global structure, namely the Markov branching property, and to know how mass is spread in one generation.”
Scaling Limits of Random Graphs from Subcritical Classes (2014, background): “See for example Duquesne [20], Duquesne and Le Gall [21], Haas and Miermont [25].”
Scaling limits of random planar maps with large faces (2011, background): “ss (n−1/αSθ [nt],n −(1−1/α)Hθ [nt])t≥0 54 J.-F. LE GALL AND G. MIERMONT under Q(n)(dθ) converges, as n→∞,to the law of the process (c0Xt,c −1 0 Ht)t≥0 under N(·|σ=1). This follows from Theorem 3.1 in [11]. This theorem gives the convergence in distribution of the rescaled height process (n−(1−1/α)Hθ”
Random Trees, Lévy Processes and Spatial Branching Processes (2005, methodology): “See however Duquesne [11] for an extension of Aldous’ theorem to the stable case using the tools of the present work (a related result in the stable case was obtained by Kersting [26]).”
Invariance principles for spatial multitype Galton–Watson trees (2006, background): “We finally stress that Proposition 2 is in fact a particular case of Duquesne and Le Gall’s results [4, 3], which deal with the case of offspring distributions belonging to other stable domains of attraction than the Gaussian one.”
Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation (2011, background): “Duquesne [34] and Le Gall [80, 81]; the limit is now a different random metric space called a stable tree.”
Sub-Gaussian tail bounds for the width and height of conditioned Galton--Watson trees (2010, background): “In particular, we ask the following questions (assuming Eξ = 1): Is EH(Tn) = O(n1/2) also if σ 2 = ∞? Is EH(Tn) = o(n1/2) if σ 2 = ∞? We note that Duquesne [13] has shown that H(Tn) = Op(n) when ξ is in the domain of attraction of a stable law of index α ∈ (1,2] [in fact, his results also imply that”
Random trees and applications (2005, background): “See also Duquesne [10] for generalisations of Theorem 1.15.”
Invariance principles for Galton-Watson trees conditioned on the number of leaves (2011, background): “[11,12]).”
Probabilistic and fractal aspects of Lévy trees (2005, background): “In a sense, the preceding result is not really new: See [2],[8] and especially Chapter 2 of [9] for related limit theorems with a different formalism.”
Random real trees (2006, background): “2 ([20]) Suppose that μ is critical and aperiodic, and that it is in the domain of attraction of the stable law with index α ∈ (1, 2), meaning that there exists a sequence an ↑ ∞ such that, if Y1, Y2, .”
Combinatorial Stochastic Processes (2006, background): “Specializing to the case where μn = μ is in the domain of attraction of a stable law, Duquesne [118] has shown that the contour process of the μ-Galton-Watson tree conditioned to have exactly n vertices converges after rescaling towards the normalized excursion of the (stable) height process, which ”
Local limits of conditioned Galton-Watson trees: the infinite spine case (2013, background): “Other limits can be considered such as scaling limits of conditioned GW trees (see [7, 18, 24]) but this is not the purpose here.”
Self-similar fragmentations derived from the stable tree II: splitting at nodes (2004, unknown): “This section is mainly inspired by [16, 14].”