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59
Coalescents With Multiple Collisions
 Ann. Probab
, 1999
"... For each finite measure on [0 ..."
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A classification of coalescent processes for haploid exchangeable population models
 Ann. Probab
, 2001
"... We consider a class of haploid population models with nonoverlapping generations and fixed population size N assuming that the family sizes within a generation are exchangeable random variables. A weak convergence criterion is established for a properly scaled ancestral process as N! 1. It results ..."
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Cited by 63 (4 self)
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We consider a class of haploid population models with nonoverlapping generations and fixed population size N assuming that the family sizes within a generation are exchangeable random variables. A weak convergence criterion is established for a properly scaled ancestral process as N! 1. It results in a full classification of the coalescent generators in the case of exchangeable reproduction. In general the coalescent process allows for simultaneous multiple mergers of ancestral lines.
Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Cited by 48 (3 self)
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
Betacoalescents and continuous stable random trees
, 2006
"... Coalescents with multiple collisions, also known as Λcoalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case t ..."
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Cited by 47 (15 self)
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Coalescents with multiple collisions, also known as Λcoalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure Λ is the Beta(2 − α, α) distribution, they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring. Here we use a recent result of Birkner et al. to prove that Betacoalescents can be embedded in continuous stable random trees, about which much is known due to recent progress of Duquesne and Le Gall. Our proof is based on a construction of the DonnellyKurtz lookdown process using continuous random trees which is of independent interest. This produces a number of results concerning the smalltime behavior of Betacoalescents. Most notably, we recover an almost sure limit theorem of the authors for the number of blocks at small times, and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the site frequency spectrum and allele frequency spectrum associated with mutations in the context of population genetics.
Random recursive trees and the BolthausenSznitman coalescent
 Electron. J. Probab
, 2005
"... We describe a representation of the BolthausenSznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the coalescent restricted to [n]: we show that the distribution of the number of blocks involved in the ..."
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Cited by 38 (2 self)
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We describe a representation of the BolthausenSznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the coalescent restricted to [n]: we show that the distribution of the number of blocks involved in the final collision converges as n → ∞, and obtain a scaling law for the sizes of these blocks. We also consider the discretetime Markov chain giving the number of blocks after each collision of the coalescent restricted to [n]; we show that the transition probabilities of the timereversal of this Markov chain have limits as n → ∞. These results can be interpreted as describing a “postgelation ” phase of the BolthausenSznitman coalescent, in which a giant cluster containing almost all of the mass has already formed and the remaining small blocks are being absorbed. 1
The contour of splitting trees is a Lévy process
, 2007
"... Splitting trees are those random trees where individuals give birth at constant rate during a lifetime with general distribution, to i.i.d. copies of themselves. The width process of a splitting tree is then a binary, homogeneous Crump–Mode–Jagers (CMJ) process, and is not Markovian unless the lifet ..."
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Cited by 36 (14 self)
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Splitting trees are those random trees where individuals give birth at constant rate during a lifetime with general distribution, to i.i.d. copies of themselves. The width process of a splitting tree is then a binary, homogeneous Crump–Mode–Jagers (CMJ) process, and is not Markovian unless the lifetime distribution is exponential (or a Dirac mass at {∞}). Here, we allow the birth rate to be infinite, that is, pairs of birth times and lifespans of newborns form a Poisson point process along the lifetime of their mother, with possibly infinite intensity measure. A splitting tree is a random (socalled) chronological tree. Each element of a chronological tree is a (socalled) existence point (v, τ) of some individual v (vertex) in a discrete tree, where τ is a nonnegative real number called chronological level (time). We introduce a total order on existence points, called linear order, and a mapping ϕ from the tree into the real line which preserves this order. The inverse of ϕ is called the exploration process, and the projection of this inverse on chronological levels the contour process. For splitting trees truncated up to level τ, we prove that thus defined contour process is a Lévy process reflected below τ and killed upon hitting 0. This allows to derive properties of (i) splitting trees: conceptual proof of Le Gall–Le Jan’s theorem in the finite variation case, exceptional points, coalescent point process, age distribution; (ii) CMJ processes: onedimensional marginals, conditionings, limit theorems, asymptotic numbers of individuals with infinite vs finite descendances. Running head. The contour of splitting trees.
The branching process with logistic growth
 Ann. Appl. Probab
, 2005
"... In order to model random densitydependence in population dynamics, we construct the random analogue of the wellknown logistic process in the branching process ’ framework. This densitydependence corresponds to intraspecific competition pressure, which is ubiquitous in ecology, and translates mathe ..."
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Cited by 31 (8 self)
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In order to model random densitydependence in population dynamics, we construct the random analogue of the wellknown logistic process in the branching process ’ framework. This densitydependence corresponds to intraspecific competition pressure, which is ubiquitous in ecology, and translates mathematically into a quadratic death rate. The logistic branching process, or LBprocess, can thus be seen as (the mass of) a fragmentation process (corresponding to the branching mechanism) combined with constant coagulation rate (the death rate is proportional to the number of possible coalescing pairs). In the continuous statespace setting, the LBprocess is a timechanged (in Lamperti’s fashion) Ornstein–Uhlenbeck type process. We obtain similar results for both constructions: when natural deaths do not occur, the LBprocess converges to a specified distribution; otherwise, it goes extinct a.s. In the latter case, we provide the expectation and the Laplace transform of the absorption time,
Asymptotic results concerning the total branch length of the Bolthausen–Sznitman coalescent
, 2007
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