Convergence of bi-measure $\mathbb{R}$-trees and the pruning process

Löhr, Wolfgang; Voisin, Guillaume; Winter, Anita

doi:10.1214/14-aihp628

Cited by 14 publications

(10 citation statements)

References 47 publications

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“…The convergence of (σ n L n t , t ≥ 0) to (L t , t ≥ 0) on any finite interval follows mainly from the convergence in Proposition 5.2. The proof here can be easily adapted to the other models of random trees, see [16,39]. On the other hand, our proof of the tightness condition (5.20) depends on the specific cuttings on the birthday trees, which has allowed us to deduce the distributional identity (5.22).…”

Section: )mentioning

confidence: 78%

“…depends continuously on the metric of T n (resp. the metric of T ), according to Proposition 2.23 of [39] this implies that, for each k ≥ 1, σ n R n k , θ 2 0 σ n n R n k → R k , θ 2 0 R k , (7.6) almost surely in the Gromov-Hausdorff-Prokhorov topology. On the other hand, we easily deduce from the convergence of the vector B n m and (H) that, for each fixed m ≥ 1,…”

Section: Distribution Of the Cutsmentioning

confidence: 95%

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Cutting down $\mathbf{p}$-trees and inhomogeneous continuum random trees

Broutin

Wang²

2017

Bernoulli

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We study a fragmentation of the p-trees of Camarri and Pitman [Elect. ] about the cut tree of the Brownian continuum random tree.associates with the cutting procedure a partial resampling of the Cayley tree of the kind mentioned earlier: if one considers the (ordered) sequence of subtrees which are discarded as the cutting process goes on, and adds a path linking their roots, then the resulting tree is a uniformly random Cayley tree, and the two extremities of the path are independent uniform random nodes. So the properties of the parameter L n follow from a stronger correspondence between the combinatorial objects themselves.This strong connection between the discrete objects can be carried to the level of their scaling limit, namely Aldous' Brownian continuum random tree (CRT) [5]. Without being too precise for now, the natural cutting procedure on the Brownian CRT involves a Poisson rain of cuts sampled according to the length measure. However, not all the cuts contribute to the isolation of the root. As in the partial resampling of the discrete setting, we glue the sequence of discarded subtrees along an interval, thereby obtaining a new CRT. If the length of the interval is well-chosen (as a function of the cutting process), the tree obtained is distributed like the Brownian CRT and the two ends of the interval are independently random leaves. This identifies the distribution of the discarded subtrees from the cutting procedure as the distribution of the forest one obtains from a spinal decomposition of the Brownian CRT. The distribution of the latter is intimately related to Bismut's [18] decomposition of a Brownian excursion. See also [25] for the generalization to the Lévy case. Note that a similar identity has been proved by Abraham and Delmas [2] for general Lévy trees without using a discrete approximation. A related example is that of the subtree prune and re-graft dynamics of Evans et al. [28] [See also 26], which is even closer to the cutting procedure and truly resamples the object rather than giving a "recursive" decomposition.The aim of this paper is two-fold. First we prove exact identities and give reversible transformations of p-trees similar to the ones for Cayley trees in [3]. The model of p-trees introduced by Camarri and Pitman [21] generalizes Cayley trees in allowing "weights" on the vertices. In particular, this additional structure of weights introduces some inhomogeneity. We then lift these results to the scaling limits, the inhomogeneous continuum random trees (ICRT) of Aldous and Pitman [8], which are closely related to the general additive coalescent [8,13,14]. Unlike the Brownian CRT or the stable trees (special cases of Lévy trees), a general ICRT is not self-similar. Nor does it enjoy a "branching property" as the Lévy trees do [37]. This lack of "recursivity" ruins the natural approaches such as the one used in [1,2] or the ones which would argue by comparing two fragmentations with the same dislocation measure but different indices of self-similarity [15]. This is one of the r...

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Section: )mentioning

confidence: 78%

Section: Distribution Of the Cutsmentioning

confidence: 95%

Cutting down $\mathbf{p}$-trees and inhomogeneous continuum random trees

Broutin

Wang²

2017

Bernoulli

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“…We consider that K 0 is equipped with the Gromov-weak topology on H n ; see Gromov's book [29] or [28,39]. In particular, the Gromov-weak topology is metrized by the so-called pointed Gromov-Prokhorov metric d pGP .…”

Section: Convergence Of the Processes Of Laminationsmentioning

confidence: 99%

“…In particular, the Gromov-weak topology is metrized by the so-called pointed Gromov-Prokhorov metric d pGP . Moreover, (K 0 , d pGP ) is a complete and separable metric space; see [39,Proposition 2.6]. Let us give a simple characterization for convergence in the Gromov-weak topology, see e.g.…”

Section: Convergence Of the Processes Of Laminationsmentioning

confidence: 99%

Convergence of trees with a given degree sequence and of their associated laminations

Ojeda¹,

Holmgren²,

Thévenin³

2021

Preprint

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In this paper, we study uniform rooted plane trees with given degree sequence. We show, under some natural hypotheses on the degree sequence, that these trees converge toward the so-called Inhomogeneous Continuum Random Tree after renormalisation. Our proof relies on the convergence of a modification of the well-known Łukasiewicz path. We also give a unified treatment of the limit, as the number of vertices tends to infinity, of the fragmentation process derived by cutting-down the edges of a tree with a given degree sequence, including its geometric representation by a lamination-valued process. The latter is a collection of nested laminations that are compact subsets of the unit disk made of non-crossing chords. In particular, we prove an equivalence between Gromov-weak convergence of discrete trees and the convergence of their associated lamination-valued processes.

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“…A version of Le Cam's theorem for separable metric spaces dropping the "Radon" assumption on the probability measures is given in [BK10]. This version was used extensively for the construction of a tree-valued pruning process in [LVW15]. Our main goal is to extend Le Cam's result to the case of boundedly finite measures and weak # -convergence and, because convergence determining is sometimes too much to ask for, to obtain (weaker) sufficient conditions for F to at least separate boundedly finite measures.…”

Section: Introductionmentioning

confidence: 99%

Boundedly finite measures: separation and convergence by an algebra of functions

Löhr¹,

Rippl²

2016

Electron. Commun. Probab.

Self Cite

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We prove general results about separation and weak # -convergence of boundedly finite measures on separable metric spaces and Souslin spaces. More precisely, we consider an algebra of bounded real-valued, or more generally a * -algebra F of bounded complex-valued functions and give conditions for it to be separating or weak # -convergence determining for those boundedly finite measures that integrate all functions in F. For separation, it is sufficient if F separates points, vanishes nowhere, and either consists of only countably many measurable functions, or of arbitrarily many continuous functions. For convergence determining, it is sufficient if F induces the topology of the underlying space, and every bounded set A admits a function in F with values bounded away from zero on A.

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Convergence of bi-measure $\mathbb{R}$-trees and the pruning process

Cited by 14 publications

References 47 publications

Cutting down $\mathbf{p}$-trees and inhomogeneous continuum random trees

Cutting down $\mathbf{p}$-trees and inhomogeneous continuum random trees

Convergence of trees with a given degree sequence and of their associated laminations

Boundedly finite measures: separation and convergence by an algebra of functions

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