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29
Poisson process partition calculus with an application to Bayesian Levy moving averages
, 2005
"... This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailormade to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The P ..."
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Cited by 57 (13 self)
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This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailormade to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The Poisson disintegration method is based on the formal statement of two results concerning a Laplace functional change of measure and a Poisson Palm/Fubini calculus in terms of random partitions of the integers {1,...,n}. The techniques are analogous to, but much more general than, techniques for the Dirichlet process and weighted gamma process developed in [Ann. Statist. 12
Martingales and Profile of Binary Search Trees
, 2004
"... We are interested in the asymptotic analysis of the binary search tree (BST) under the random permutation model. Via an embedding in a continuous time model, we get new results, in particular the asymptotic behavior of the profile. ..."
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Cited by 48 (11 self)
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We are interested in the asymptotic analysis of the binary search tree (BST) under the random permutation model. Via an embedding in a continuous time model, we get new results, in particular the asymptotic behavior of the profile.
Selfsimilar fragmentations
, 2000
"... We introduce a probabilistic model that is meant to describe an object that falls apart randomly as time passes and fulfills a certain scaling property. We show that the distribution of such a process is determined by its index of selfsimilarity α ∈ R, a rate of erosion c ≥ 0, and a socalled Lév ..."
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Cited by 37 (9 self)
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We introduce a probabilistic model that is meant to describe an object that falls apart randomly as time passes and fulfills a certain scaling property. We show that the distribution of such a process is determined by its index of selfsimilarity α ∈ R, a rate of erosion c ≥ 0, and a socalled Lévy measure that accounts for sudden dislocations. The key of the analysis is provided by a transformation of selfsimilar fragmentations which enables us to reduce the study to the homogeneous case α = 0 which is treated in [6].
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.
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 probabilistic analysis of some tree algorithms
 ANNALS OF APPLIED PROBABILITY
, 2005
"... In this paper a general class of tree algorithms is analyzed. It is shown that, by using an appropriate probabilistic representation of the quantities of interest, the asymptotic behavior of these algorithms can be obtained quite easily without resorting to the usual complex analysis techniques. Thi ..."
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Cited by 22 (6 self)
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In this paper a general class of tree algorithms is analyzed. It is shown that, by using an appropriate probabilistic representation of the quantities of interest, the asymptotic behavior of these algorithms can be obtained quite easily without resorting to the usual complex analysis techniques. This approach gives a unified probabilistic treatment of these questions. It simplifies and extends some of the results known in this domain.
Gibbs Fragmentation Trees
, 2008
"... We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbstype fragmentation tree with Aldous’ betasplitting model, which has an extended parameter range β>−2 with respect to the beta(β + 1,β + 1) probability distributions on which it is based. In the mul ..."
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Cited by 19 (6 self)
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We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbstype fragmentation tree with Aldous’ betasplitting model, which has an extended parameter range β>−2 with respect to the beta(β + 1,β + 1) probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the twoparameter Poisson–Dirichlet models for exchangeable random partitions of N, with an extended parameter range 0 ≤ α ≤ 1, θ ≥−2α and α<0, θ =−mα, m ∈ N.
Double Martingale Structure and Existence of φMoments for Weighted Branching Processes
, 2007
"... Given the infinite UlamHarris tree V = ∪n≥0Nn, let T (v) =(Ti(v))i≥1, v ∈ V, be a familiy of i.i.d. nonnegative random vectors with generic copy (Ti)i≥1. Interpret Ti(v) as a weight attached to the edge connecting the nodes v and vi in the tree. Define L(v) asthe branch weight for the unique path f ..."
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Cited by 14 (6 self)
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Given the infinite UlamHarris tree V = ∪n≥0Nn, let T (v) =(Ti(v))i≥1, v ∈ V, be a familiy of i.i.d. nonnegative random vectors with generic copy (Ti)i≥1. Interpret Ti(v) as a weight attached to the edge connecting the nodes v and vi in the tree. Define L(v) asthe branch weight for the unique path from the root to v obtained by multiplication of the edge weights. The associated weighted branching process (WBP) def ∑ is then given by Zn = L(v), n ≥ 0, and forms a nonnegav=n ∑tive martingale with a.s. limit W under the normalization assumption i≥1 ETi =1. For regularly varying functions φ(x) =xαℓ(x) of order α ≥ 1 satisfying lim x→ ∞ x−1φ(x) =∞, the paper provides necessary and sufficient conditions on (Ti)i≥1 for Eφ(W)being positive and finite. The double martingale structure of (Zn)n≥0 first observed and utilized in [5] for similar results for GaltonWatson processes forms a major tool in our analysis. It further requires results following from the connection between a WBP and an associated random walk and drawing on results from renewal theory. In particular, a pathwise renewal theorem is proved which may also be of interest in its own right.
Exchangeable fragmentationcoalescence processes and their equilibrium distribution
 Electr. J. Prob
, 2004
"... We define and study a family of Markov processes with state space the compact set of all partitions of N that we call exchangeable fragmentationcoalescence processes. They can be viewed as a combination of exchangeable fragmentation as defined by Bertoin and of homogenous coalescence as defined by ..."
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Cited by 13 (1 self)
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We define and study a family of Markov processes with state space the compact set of all partitions of N that we call exchangeable fragmentationcoalescence processes. They can be viewed as a combination of exchangeable fragmentation as defined by Bertoin and of homogenous coalescence as defined by Pitman and Schweinsberg or Möhle and Sagitov. We show that they admit a unique invariant probability measure and we study some properties of their paths and of their equilibrium measure. Key words. Fragmentation, coalescence, invariant distribution. A.M.S. Classification. 60 J 25, 60 G 09. 1
On small masses in selfsimilar fragmentations
, 2004
"... We consider a selfsimilar fragmentation process which preserves the total mass. We are interested in the asymptotic behavior as ε → 0+ of N ( ; t) =Card{i: Xi(t) ¿ε}, the number of fragments with size greater than at some fixed time t¿0. Under a certain condition of regular variation type on the s ..."
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Cited by 11 (0 self)
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We consider a selfsimilar fragmentation process which preserves the total mass. We are interested in the asymptotic behavior as ε → 0+ of N ( ; t) =Card{i: Xi(t) ¿ε}, the number of fragments with size greater than at some fixed time t¿0. Under a certain condition of regular variation type on the socalled dislocation measure, we exhibit a deterministic function ’:]0; 1 [ →]0; ∞ [ such that the limit of N ( ; t)=’ ( ) exists and is nondegenerate. In general the limit is random, but may be deterministic when a certain relation between the index of selfsimilarity and the dislocation measure holds. We also present a similar result for the total mass of fragments less than ε.