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22
PRUNING A LÉVY CONTINUUM RANDOM TREE
, 804
"... Abstract. Given a general critical or subcritical branching mechanism, we define a pruning procedure of the associated Lévy continuum random tree. This pruning procedure is defined by adding some marks on the tree, using Lévy snake techniques. We then prove that the resulting subtree after pruning ..."
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Cited by 18 (11 self)
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Abstract. Given a general critical or subcritical branching mechanism, we define a pruning procedure of the associated Lévy continuum random tree. This pruning procedure is defined by adding some marks on the tree, using Lévy snake techniques. We then prove that the resulting subtree after pruning is still a Lévy continuum random tree. This last result is proved using the exploration process that codes the CRT, a special Markov property and martingale problems for exploration processes. We finally give the joint law under the excursion measure of the lengths of the excursions of the initial exploration process and the pruned one. 1.
Exit times for an increasing Lévy treevalued process
, 2012
"... We give an explicit construction of the increasing treevalued process introduced by Abraham and Delmas using a random point process of trees and a grafting procedure. This random point process will be used in companion papers to study record processes on Lévy trees. We use the Poissonian structure ..."
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Cited by 16 (9 self)
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We give an explicit construction of the increasing treevalued process introduced by Abraham and Delmas using a random point process of trees and a grafting procedure. This random point process will be used in companion papers to study record processes on Lévy trees. We use the Poissonian structure of the jumps of the increasing treevalued process to describe its behavior at the first time the tree grows higher than a given height. We also give the joint distribution of this exit time and the ascension time which corresponds to the first infinite jump of the treevalued process.
Pruning Galton–Watson trees and treevalued Markov processes. Preprint. Available at arXiv:1007.0370
, 2010
"... Abstract. We present a new pruning procedure on discrete trees by adding marks on the nodes of trees. This procedure allows us to construct and study a treevalued Markov process {G(u)} by pruning GaltonWatson trees and an analogous process {G ∗ (u)} by pruning a critical or subcritical GaltonWats ..."
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Cited by 14 (12 self)
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Abstract. We present a new pruning procedure on discrete trees by adding marks on the nodes of trees. This procedure allows us to construct and study a treevalued Markov process {G(u)} by pruning GaltonWatson trees and an analogous process {G ∗ (u)} by pruning a critical or subcritical GaltonWatson tree conditioned tobe infinite. Underamild condition on offspring distributions, we show that the process {G(u)} run until its ascension time has a representation in terms of {G ∗ (u)}. A similar result was obtained by Aldous and Pitman (1998) in the special case of Poisson offspring distributions where they considered uniform pruning of GaltonWatson trees by adding marks on the edges of trees. hal00497035, version 2 7 Feb 2011 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.
Smaller population size at the MRCA time for stationary branching processes
, 2010
"... We present an elementary model of random size varying population given by a stationary continuous state branching process. For this model we compute the joint distribution of: the time to the most recent common ancestor, the size of the current population and the size of the population just before ..."
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Cited by 11 (7 self)
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We present an elementary model of random size varying population given by a stationary continuous state branching process. For this model we compute the joint distribution of: the time to the most recent common ancestor, the size of the current population and the size of the population just before the most recent common ancestor (MRCA). In particular we show a natural mild bottleneck effect as the size of the population just before the MRCA is stochastically smaller than the size of the current population. We also compute the number of old families which corresponds to the number of individuals involved in the last coalescent event of the genealogical tree. By studying more precisely the genealogical structure of the population, we get asymptotics for the number of ancestors just before the current time. We give explicit computations in the case of the quadratic branching mechanism. In this case, the size of the population at the MRCA is, in mean, less by 1/3 than size of the current population size. We also provide in this case the fluctuations for the renormalized number of ancestors.
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
A Williams decomposition for spatially dependent superprocesses ∗
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A CONSTRUCTION OF A βCOALESCENT VIA THE PRUNING OF BINARY TREES
"... Abstract. Considering a random binary tree with n labelled leaves, we use a pruning procedure on this tree in order to construct a β ( 3 1,)coalescent process. We also use the 2 2 continuous analogue of this construction, i.e. a pruning procedure on Aldous’s continuum random tree, to construct a co ..."
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Cited by 3 (1 self)
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Abstract. Considering a random binary tree with n labelled leaves, we use a pruning procedure on this tree in order to construct a β ( 3 1,)coalescent process. We also use the 2 2 continuous analogue of this construction, i.e. a pruning procedure on Aldous’s continuum random tree, to construct a continuous state space process that has the same structure as the βcoalescent process up to some time change. These two constructions enable us to obtain results on the coalescent process such as the asymptotics on the number of coalescent events or the law of the blocks involved in the last coalescent event. hal00711518, version 2 9 Nov 2012 1.