Results 1  10
of
14
Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws
, 2008
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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 26 (11 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.
Asymptotic laws for compositions derived from transformed subordinators
 ANN. PROBAB
, 2006
"... A random composition of n appears when the points of a random closed set ˜ R ⊂ [0, 1] are used to separate into blocks n points sampled from the uniform distribution. We study the number of parts Kn of this composition and other related functionals under the assumption that ˜ R = φ(S•) where (St, t ..."
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Cited by 24 (10 self)
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A random composition of n appears when the points of a random closed set ˜ R ⊂ [0, 1] are used to separate into blocks n points sampled from the uniform distribution. We study the number of parts Kn of this composition and other related functionals under the assumption that ˜ R = φ(S•) where (St, t ≥ 0) is a subordinator and φ: [0, ∞] → [0, 1] is a diffeomorphism. We derive the asymptotics of Kn when the Lévy measure of the subordinator is regularly varying at 0 with positive index. Specialising to the case of exponential function φ(x) = 1 −e −x we establish a connection between the asymptotics of Kn and the exponential functional of the subordinator.
Exchangeable partitions derived from Markovian coalescents
 Adv. Appl. Probab
, 2006
"... Kingman derived the Ewens sampling formula for random partitions from the genealogy model defined by a Poisson process of mutations along lines of descent governed by a simple coalescent process. Möhle described the recursion which determines the generalization of the Ewens sampling formula when the ..."
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Cited by 14 (3 self)
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Kingman derived the Ewens sampling formula for random partitions from the genealogy model defined by a Poisson process of mutations along lines of descent governed by a simple coalescent process. Möhle described the recursion which determines the generalization of the Ewens sampling formula when the lines of descent are governed by a coalescent with multiple collisions. In [7] authors exploit an analogy with the theory of regenerative composition and partition structures, and provide various characterizations of the associated exchangeable random partitions. This paper gives parallel results for the further generalized model with lines of descent following a coalescent with simultaneous multiple collisions. 1
Selfsimilar and Markov compositions structures
 Metody
, 2005
"... Abstract The bijection between composition structures and random closed subsets of the unit interval implies that the composition structures associated with S ∩[0, 1] for a selfsimilar random set S ⊂ R+ are those which are consistent with respect to a simple truncation operation. Using the standard ..."
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Cited by 9 (5 self)
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Abstract The bijection between composition structures and random closed subsets of the unit interval implies that the composition structures associated with S ∩[0, 1] for a selfsimilar random set S ⊂ R+ are those which are consistent with respect to a simple truncation operation. Using the standard coding of compositions by finite strings of binary digits starting with a 1, the random composition of n is defined by the first n terms of a random binary sequence of infinite length. The locations of 1s in the sequence are the places visited by an increasing timehomogeneous Markov chain on the positive integers if and only if S = exp(−W) for some stationary regenerative random subset W of the real line. Complementing our study in previous papers, we identify selfsimilar Markovian composition structures associated with the twoparameter family of partition structures. 1
Asymptotics of the allele frequency spectrum associated with the BolthausenSznitman coalescent
, 2007
"... We work in the context of the infinitely many alleles model. The allelic partition associated with a coalescent process started from n individuals is obtained by placing mutations along the skeleton of the coalescent tree; for each individual, we trace back to the most recent mutation affecting it a ..."
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Cited by 8 (0 self)
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We work in the context of the infinitely many alleles model. The allelic partition associated with a coalescent process started from n individuals is obtained by placing mutations along the skeleton of the coalescent tree; for each individual, we trace back to the most recent mutation affecting it and group together individuals whose most recent mutations are the same. The number of blocks of each of the different possible sizes in this partition is the allele frequency spectrum. The celebrated Ewens sampling formula gives precise probabilities for the allele frequency spectrum associated with Kingman’s coalescent. This (and the degenerate starshaped coalescent) are the only Λcoalescents for which explicit probabilities are known, although they are known to satisfy a recursion due to Möhle. Recently, Berestycki, Berestycki and Schweinsberg have proved asymptotic results for the allele frequency spectra of the Beta(2 − α,α) coalescents with α ∈ (1,2). In this paper, we prove full asymptotics for the case of the BolthausenSznitman coalescent.
Regenerative compositions in the case of slow variation
 Stoch. Process. Appl
, 2006
"... For S a subordinator and Πn an independent Poisson process of intensity ne −x,x> 0, we are interested in the number Kn of gaps in the range of S that are hit by at least one point of Πn. Extending previous studies in [7, 10, 11] we focus on the case when the tail of the Lévy measure of S is slowl ..."
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Cited by 7 (4 self)
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For S a subordinator and Πn an independent Poisson process of intensity ne −x,x> 0, we are interested in the number Kn of gaps in the range of S that are hit by at least one point of Πn. Extending previous studies in [7, 10, 11] we focus on the case when the tail of the Lévy measure of S is slowly varying. We view Kn as the terminal value of a random process Kn, and provide an asymptotic analysis of the fluctuations of Kn, as n → ∞, for a wide spectrum of situations. 1
Moments of convex distribution functions and completely alternating sequences
 In Festschrift for Avner Friedman (IMS Lecture
, 2006
"... Abstract: We solve the moment problem for convex distribution functions on [0,1] in terms of completely alternating sequences. This complements a recent solution of this problem by Diaconis and Freedman, and relates this work to the LévyKhintchine formula for the Laplace transform of a subordinator ..."
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Cited by 7 (4 self)
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Abstract: We solve the moment problem for convex distribution functions on [0,1] in terms of completely alternating sequences. This complements a recent solution of this problem by Diaconis and Freedman, and relates this work to the LévyKhintchine formula for the Laplace transform of a subordinator, and to regenerative composition structures. 1.
A twoparameter family infinitedimensional diffusions in the Kingman simplex
, 2007
"... The main result of the present paper is to construct a twoparameter family of Markov processes Xα,θ(t) in the infinitedimensional Kingman simplex ..."
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Cited by 6 (1 self)
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The main result of the present paper is to construct a twoparameter family of Markov processes Xα,θ(t) in the infinitedimensional Kingman simplex
Poisson representation of a Ewens fragmentation process
"... A simple explicit construction is provided of a partitionvalued fragmentation process whose distribution on partitions of [n] = {1,...,n} at time θ ≥ 0 is governed by the Ewens sampling formula with parameter θ. These partitionvalued processes are exchangeable and consistent, as n varies. They ca ..."
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Cited by 6 (3 self)
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A simple explicit construction is provided of a partitionvalued fragmentation process whose distribution on partitions of [n] = {1,...,n} at time θ ≥ 0 is governed by the Ewens sampling formula with parameter θ. These partitionvalued processes are exchangeable and consistent, as n varies. They can be derived by uniform sampling from a corresponding mass fragmentation process defined by cutting a unit interval at the points of a Poisson process with intensity θx −1 dx on R+, arranged to be intensifying as θ increases. 1