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37
Exponential functionals of Lévy processes
 Probabilty Surveys
, 2005
"... Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0 ..."
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Cited by 76 (6 self)
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Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0
Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models
 Ann. Probab
, 2008
"... Given any regularly varying dislocation measure, we identify a natural selfsimilar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous’s betasplitting models and Ford’s alpha models for phyl ..."
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Cited by 37 (13 self)
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Given any regularly varying dislocation measure, we identify a natural selfsimilar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous’s betasplitting models and Ford’s alpha models for phylogenetic trees. This confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf.
Homogeneous fragmentation processes
, 2000
"... The purpose of this work is to define and study homogeneous fragmentation processes in continuous time, which are meant to describe the evolution of an object that breaks down randomly into pieces as time passes. Roughly, we show that the dynamic of such a fragmentation process is determined by som ..."
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Cited by 29 (4 self)
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The purpose of this work is to define and study homogeneous fragmentation processes in continuous time, which are meant to describe the evolution of an object that breaks down randomly into pieces as time passes. Roughly, we show that the dynamic of such a fragmentation process is determined by some exchangeable measure on the set of partitions of N, and results from the combination of two different phenomena: a continuous erosion and sudden dislocations. In particular, we determine the class of fragmentation measures which can arise in this setting, and investigate the evolution of the size of the fragment that contains a point pick at random at the initial time.
Asymptotic laws for nonconservative selfsimilar fragmentations
, 2008
"... We consider a selfsimilar fragmentation process in which the generic particle of size x is replaced at probability rate x α by its offspring made of smaller particles, where α is some positive parameter. The total of offspring sizes may be both larger or smaller than x with positive probability. W ..."
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Cited by 27 (4 self)
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We consider a selfsimilar fragmentation process in which the generic particle of size x is replaced at probability rate x α by its offspring made of smaller particles, where α is some positive parameter. The total of offspring sizes may be both larger or smaller than x with positive probability. We show that under certain conditions the typical size in the ensemble is of the order t −1/α and that the empirical distribution of sizes converges to a random limit which we characterise in terms of the reproduction law.
SPINAL PARTITIONS AND INVARIANCE UNDER REROOTING OF CONTINUUM RANDOM TREES
, 2009
"... We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a twoparameter Poisson–Dirichlet family of continuous fragmentation trees ..."
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Cited by 23 (12 self)
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We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a twoparameter Poisson–Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees are the only ones whose distribution is invariant under uniform rerooting.
A continuumtreevalued Markov process
"... Abstract. We present a construction of a Lévy continuum random tree (CRT) associated with a supercritical continuous state branching process using the socalled exploration process and a Girsanov’s theorem. We also extend the pruning procedure to this supercritical case. Let ψ be a critical branchi ..."
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Cited by 22 (12 self)
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Abstract. We present a construction of a Lévy continuum random tree (CRT) associated with a supercritical continuous state branching process using the socalled exploration process and a Girsanov’s theorem. We also extend the pruning procedure to this supercritical case. Let ψ be a critical branching mechanism. We set ψθ(·) = ψ( · + θ) − ψ(θ). Let Θ = (θ∞,+∞) or Θ = [θ∞,+∞) be the set of values of θ for which ψθ is a branching mechanism. The pruning procedure allows to construct a decreasing LévyCRTvalued Markov process (Tθ, θ ∈ Θ), such that Tθ has branching mechanism ψθ. It is subcritical if θ> 0 and supercritical if θ < 0. We then consider the explosion time A of the CRT: the smaller (negative) time θ for which Tθ has finite mass. We describe the law of A as well as the distribution of the CRT just after this explosion time. The CRT just after explosion can be seen as a CRT conditioned not to be extinct which is pruned with an independent intensity related to A. We also study the evolution of the CRTvalued process after the explosion time. This extends results from Aldous and Pitman on GaltonWatson trees. For the particular case of the quadratic branching mechanism, we show that after explosion the total mass of the CRT behaves like the inverse of a stable subordinator with index 1/2. This result is related to the size of the tagged fragment for the fragmentation of Aldous ’ CRT. 1.
Coagulationfragmentation duality, Poisson–Dirichlet distributions and random recursive trees
 Ann. Appl. Probab
, 2006
"... In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the twoparameter family of Poisson–Dirichlet distributions PD(α, θ) that take values in ..."
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In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the twoparameter family of Poisson–Dirichlet distributions PD(α, θ) that take values in this space. We introduce families of random fragmentation and coagulation operators Frag α and Coag α,θ, respectively, with the following property: if the input to Frag α has PD(α, θ) distribution, then the output has PD(α, θ +1) distribution, while the reverse is true for Coag α,θ. This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between PD(α, θ) and PD(αβ, θ). Repeated application of the Frag α operators gives rise to a family of fragmentation chains. We show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coagulation–fragmentation duality. 1. Introduction. The
Regenerative tree growth: binary selfsimilar continuum random trees and PoissonDirichlet compositions
, 2009
"... We use a natural ordered extension of the Chinese Restaurant Process to grow a twoparameter family of binary selfsimilar continuum fragmentation trees. We provide an explicit embedding of Ford’s sequence of alpha model trees in the continuum tree which we identified in a previous article as a dist ..."
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Cited by 16 (6 self)
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We use a natural ordered extension of the Chinese Restaurant Process to grow a twoparameter family of binary selfsimilar continuum fragmentation trees. We provide an explicit embedding of Ford’s sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of Ford’s trees. In general, the Markov branching trees induced by the twoparameter growth rule are not sampling consistent, so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case. We develop here a new approach to establish such limits, based on regenerative interval partitions and the urnmodel description of sampling from Dirichlet random distributions.
Connecting yule process, bisection and binary search tree via martingales
 JIRSS
"... Abstract. We present new links between some remarkable martingales found in the study of the Binary Search Tree or of the bisection problem, looking at them on the probability space of a continuous time binary branching process. 1 ..."
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Cited by 13 (4 self)
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Abstract. We present new links between some remarkable martingales found in the study of the Binary Search Tree or of the bisection problem, looking at them on the probability space of a continuous time binary branching process. 1