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23
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 26 (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.
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 24 (8 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].
Inhomogeneous Continuum Random Trees and the Entrance Boundary of the Additive Coalescent
 PROBAB. TH. REL. FIELDS
, 1998
"... Regard an element of the set of ranked discrete distributions \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; P i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent is the \Deltavalued Markov process in which pairs of clusters of masses fx i ; ..."
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Cited by 18 (13 self)
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Regard an element of the set of ranked discrete distributions \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; P i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent is the \Deltavalued Markov process in which pairs of clusters of masses fx i ; x j g merge into a cluster of mass x i + x j at rate x i + x j . Aldous and Pitman (1998) showed that a version of this process starting from time \Gamma1 with infinitesimally small clusters can be constructed from the Brownian continuum random tree of Aldous (1991,1993) by Poisson splitting along the skeleton of the tree. In this paper it is shown that the general such process may be constructed analogously from a new family of inhomogeneous continuum random trees.
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
Asymptotic laws for nonconservative selfsimilar fragmentations
 Electronic J. Probab
, 2004
"... Abstract 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 probab ..."
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Cited by 14 (2 self)
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Abstract 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. 1
Two coalescents derived from the ranges of stable subordinators
 Electron. J. Probab
"... E l e c t r o n ..."
Asymptotic distributions for the cost of linear probing hashing, Random Structures and Algorithms
"... Abstract. We study moments and asymptotic distributions of the construction cost, measured as the total displacement, for hash tables using linear probing. Four different methods are employed for different ranges of the parameters; together they yield a complete description. This extends earlier res ..."
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Cited by 11 (3 self)
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Abstract. We study moments and asymptotic distributions of the construction cost, measured as the total displacement, for hash tables using linear probing. Four different methods are employed for different ranges of the parameters; together they yield a complete description. This extends earlier results by Flajolet, Poblete and Viola. The average cost of unsuccessful searches is considered too. 1.
PoissonKingman Partitions
 of Lecture NotesMonograph Series
, 2002
"... This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are the points of a Poisson point process. These lengths can be also interpreted as the jumps of a subordin ..."
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Cited by 11 (3 self)
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This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are the points of a Poisson point process. These lengths can be also interpreted as the jumps of a subordinator, that is an increasing process with stationary independent increments. Examples include the twoparameter family of PoissonDirichlet models derived from the Poisson process of jumps of a stable subordinator. Applications are made to the random partition generated by the lengths of excursions of a Brownian motion or Brownian bridge conditioned on its local time at zero.