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11
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 27 (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.
The structure of the allelic partition of the total population for GaltonWatson processes with neutral mutations
"... We consider a (sub)critical Galton–Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clonechildren and ..."
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Cited by 15 (3 self)
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We consider a (sub)critical Galton–Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clonechildren and the number of mutantchildren of a typical individual. The approach combines an extension of Harris representation of Galton–Watson processes and a version of the ballot theorem. Some limit theorems related to the distribution of the allelic partition are also given. 1. Introduction. We consider a Galton–Watson process, that is, a population model with asexual reproduction such that at every generation, each individual gives birth to a random number of children according to a fixed distribution and independently of the other individuals in the population. We are interested in the situation where a child can be either a clone, that
Ranked fragmentations
 ESAIM P&S
"... distributions for random partitions generated by a ..."
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Cited by 13 (3 self)
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distributions for random partitions generated by a
On the number of collisions in Λcoalescents
 ELECTRON. J. PROBAB
, 2007
"... We examine the total number of collisions Cn in the Λcoalescent process which starts with n particles. A linear growth and a stable limit law for Cn are shown under the assumption of a powerlike behaviour of the measure Λ near 0 with exponent 0 < α < 1. ..."
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Cited by 10 (1 self)
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We examine the total number of collisions Cn in the Λcoalescent process which starts with n particles. A linear growth and a stable limit law for Cn are shown under the assumption of a powerlike behaviour of the measure Λ near 0 with exponent 0 < α < 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 7 (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.
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 5 (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
Twoparameter family of diffusion processes in the Kingman simplex, arXiv: math.PR/0708.1930
, 2007
"... §1. The Kingman graph §2. Symmetric algebra §3. Symmetric functions and the Kingman simplex ..."
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Cited by 2 (0 self)
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§1. The Kingman graph §2. Symmetric algebra §3. Symmetric functions and the Kingman simplex
Asymptotics of the allele frequency spectrum associated with the BolthausenSznitman coalescent
"... (joint work with AnneLaure Basdevant) We imagine a coalescent process as modelling the genealogy of a sample from a population which is subject to neutral mutation. We work under the assumptions of the infinitely many alleles model so that, in particular, every mutation gives rise to a completely n ..."
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Cited by 2 (0 self)
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(joint work with AnneLaure Basdevant) We imagine a coalescent process as modelling the genealogy of a sample from a population which is subject to neutral mutation. We work under the assumptions of the infinitely many alleles model so that, in particular, every mutation gives rise to a completely new type in the population. Mutations occur as a Poisson process of rate ρ along the branches of the coalescent tree. The allelic partition groups together individuals of the same allelic type, and is obtained by tracing each individual’s lineage back in time to the most recent mutation. An example of this construction is given below. 1 7
Regeneration in Random Combinatorial Structures
, 2009
"... Theory of Kingman’s partition structures has two culminating points • the general paintbox representation, relating finite partitions to hypothetical infinite populations via a natural sampling procedure, • a central example of the theory: the EwensPitman twoparameter partitions. In these notes we ..."
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Cited by 2 (2 self)
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Theory of Kingman’s partition structures has two culminating points • the general paintbox representation, relating finite partitions to hypothetical infinite populations via a natural sampling procedure, • a central example of the theory: the EwensPitman twoparameter partitions. In these notes we further develop the theory by • passing to structures enriched by the order on the collection of categories, • extending the class of tractable models by exploring the idea of regeneration, • analysing regenerative properties of the EwensPitman partitions, • studying asymptotic features of the regenerative compositions.
unknown title
, 2006
"... Summary. A homogeneous massfragmentation, as it has been defined in [6], describes the evolution of the collection of masses of fragments of an object which breaks down into pieces as time passes. Here, we show that this model can be enriched by considering also the types of the fragments, where a ..."
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Summary. A homogeneous massfragmentation, as it has been defined in [6], describes the evolution of the collection of masses of fragments of an object which breaks down into pieces as time passes. Here, we show that this model can be enriched by considering also the types of the fragments, where a type may represent, for instance, a geometrical shape, and can take finitely many values. In this setting, the dynamics of a randomly tagged fragment play a crucial role in the analysis of the fragmentation. They are determined by a Markov additive process whose distribution depends explicitly on the characteristics of the fragmentation. As applications, we make explicit the connexion with multitype branching random walks, and obtain multitype analogs of the pathwise central limit theorem and large deviation estimates for the empirical distribution of fragments. Key words. Multitype, fragmentation, branching process, Markov additive process. A.M.S. Classification. 60J80, 60G18