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16
A note on GromovHausdorffProkhorov distance between (locally) compact measure spaces
, 2012
"... Abstract. We present an extension of the GromovHausdorff metric on the set of compact metric spaces: the GromovHausdorffProkhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the noncompact case by describing a metric on the set of rooted complet ..."
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Cited by 22 (4 self)
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Abstract. We present an extension of the GromovHausdorff metric on the set of compact metric spaces: the GromovHausdorffProkhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the noncompact case by describing a metric on the set of rooted complete locally compact length spaces endowed with a locally finite measure. We prove that this space with the extended GromovHausdorffProkhorov metric is a Polish space. This generalization is needed to define Lévy trees, which are (possibly unbounded) random real trees endowed with a locally finite measure. hal00673921, version 1 24 Feb 2012 1.
Record process on the Continuum Random Tree. Arxiv preprint arXiv:1107.3657
, 2011
"... Abstract. By considering a continuous pruning procedure on Aldous’s Brownian tree, we construct a random variable Θ which is distributed, conditionally given the tree, according to the probability law introduced by Janson as the limit distribution of the number of cuts needed to isolate the root in ..."
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Cited by 12 (5 self)
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Abstract. By considering a continuous pruning procedure on Aldous’s Brownian tree, we construct a random variable Θ which is distributed, conditionally given the tree, according to the probability law introduced by Janson as the limit distribution of the number of cuts needed to isolate the root in a critical GaltonWatson tree. We also prove that this random variable can be obtained as the a.s. limit of the number of cuts needed to cut down the subtree of the continuum tree spanned by n leaves.
The forest associated with the record process on a Lévy tree
, 2012
"... Abstract. We perform a pruning procedure on a Lévy tree and instead of throwing away theremovedsubtree, we regraft itonagivenbranch(notrelated totheLévytree). Weprove that the tree constructed by regrafting is distributed as the original Lévy tree, generalizing a result of AddarioBerry, Broutin an ..."
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Cited by 7 (3 self)
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Abstract. We perform a pruning procedure on a Lévy tree and instead of throwing away theremovedsubtree, we regraft itonagivenbranch(notrelated totheLévytree). Weprove that the tree constructed by regrafting is distributed as the original Lévy tree, generalizing a result of AddarioBerry, Broutin and Holmgren where only Aldous’s tree is considered. As a consequence, we obtain that the “average pruning time ” of a leaf is distributed as the height of a leaf picked at random in the Lévy tree. hal00686569, version 2 17 Dec 2012 1.
CONVERGENCE OF BIMEASURE RTREES AND THE PRUNING PROCESS
, 2013
"... In [AP98b] a treevalued Markov chain is derived by pruning off more and more subtrees along the edges of a GaltonWatson tree. More recently, in [AD12], a continuous analogue of the treevalued pruning dynamics is constructed along Lévy trees. In the present paper, we provide a new topology which a ..."
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Cited by 5 (2 self)
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In [AP98b] a treevalued Markov chain is derived by pruning off more and more subtrees along the edges of a GaltonWatson tree. More recently, in [AD12], a continuous analogue of the treevalued pruning dynamics is constructed along Lévy trees. In the present paper, we provide a new topology which allows to link the discrete and the continuous dynamics by considering them as instances of the same strong Markov process with different initial conditions. We construct this pruning process on the space of socalled bimeasure trees, which are metric measure spaces with an additional pruning measure. The pruning measure is assumed to be finite on finite trees, but not necessarily locally finite. We also characterize the pruning process analytically via its Markovian generator and show that it is continuous in the initial bimeasure tree. A series of examples is given, which include the finite variance offspring case where the pruning measure is the length measure on the underlying tree. Résumé
Equivalence of GromovProhorov and Gromov’s ✷ λmetric on the space of metric measure spaces
 ELECTRONIC COMMUNICATIONS IN PROBABILITY
, 2013
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βCOALESCENTS AND STABLE GALTONWATSON TREES
, 2013
"... Abstract. Representation of coalescent process using pruning of trees has been used by Goldschmidt and Martin for the BolthausenSznitman coalescent and by Abraham and Delmas for the β(3/2,1/2)coalescent. By considering a pruning procedure on stable GaltonWatson tree with n labeled leaves, we give ..."
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Cited by 1 (0 self)
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Abstract. Representation of coalescent process using pruning of trees has been used by Goldschmidt and Martin for the BolthausenSznitman coalescent and by Abraham and Delmas for the β(3/2,1/2)coalescent. By considering a pruning procedure on stable GaltonWatson tree with n labeled leaves, we give a representation of the discrete β(1+α,1−α)coalescent, with α ∈ [1/2,1) starting from the trivial partition of the n first integers. The construction can also be made directly on the stable continuum Lévy tree, with parameter 1/α, simultaneously for all n. This representation allows to use results on the asymptotic number of coalescence events to get the asymptotic number of cuts in stable GaltonWatson tree (with infinite variance for the reproduction law) needed to isolate the root. Using convergence ofthestable GaltonWatson tree conditioned tohaveinfinitelymanyleaves, one can get the asymptotic distribution of blocks in the last coalescence event in the β(1+α,1−α)coalescent. 1.
Pruning of CRT subtrees
 In preparation
, 2012
"... Abstract. We study the pruning process developed by Abraham and Delmas (2012) on the discrete GaltonWatson subtrees of the Lévy tree which are obtained by considering the minimal subtree connecting the root and leaves chosen uniformly at rate λ, see Duquesne and Le Gall (2002). The treevalued pr ..."
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Abstract. We study the pruning process developed by Abraham and Delmas (2012) on the discrete GaltonWatson subtrees of the Lévy tree which are obtained by considering the minimal subtree connecting the root and leaves chosen uniformly at rate λ, see Duquesne and Le Gall (2002). The treevalued process, as λ increases, has been studied by Duquesne and Winkel (2007). Notice that we have a treevalued process indexed by two parameters the pruning parameter θ and the intensity λ. Our main results are: construction and marginals of the pruning process, representation of the pruning process (forward in time that is as θ increases) and description of the growing process (backward in time that is as θ decreases) and distribution of the ascension time (or explosion time of the backward process) as well as the tree at the ascension time. A byproduct of our result is that the supercritical Lévy trees independently introduced by Abraham and Delmas (2012) and Duquesne and Winkel (2007) coincide. This work is also related to the pruning of discrete GaltonWatson trees studied by Abraham, Delmas and He (2012). 1.