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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 46 (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.
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 29 (4 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
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 15 (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 14 (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.
The asymptotic distribution of the length of betacoalescent trees
 Ann. Appl. Probab
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Selfsimilar scaling limits of nonincreasing Markov chains
, 2009
"... We study scaling limits of nonincreasing Markov chains with values in the set of nonnegative integers, under the assumption that the large jump events are rare and happen at rates that behave like a negative power of the current state. We show that the chain starting from n and appropriately resca ..."
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Cited by 11 (1 self)
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We study scaling limits of nonincreasing Markov chains with values in the set of nonnegative integers, under the assumption that the large jump events are rare and happen at rates that behave like a negative power of the current state. We show that the chain starting from n and appropriately rescaled, converges in distribution, as n → ∞, to a nonincreasing selfsimilar Markov process. This convergence holds jointly with that of the rescaled absorption time to the time at which the selfsimilar Markov process reaches first 0. We discuss various applications to the study of random walks with a barrier, of the number of collisions in Λcoalescents that do not descend from infinity and of nonconsistent regenerative compositions. Further applications to the scaling limits of Markov branching trees are developed in the forthcoming paper [11].
On the length of an external branch in the betacoalescents
, 2012
"... In this paper, we consider Beta(2 − α,α) (with 1 < α < 2) and related Λcoalescents. If T (n) denotes the length of an external branch of the ncoalescent, we prove the convergence of n α−1 T (n) when n tends to ∞, and give the limit. To this aim, we give asymptotics for the number σ (n) of ..."
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Cited by 4 (3 self)
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In this paper, we consider Beta(2 − α,α) (with 1 < α < 2) and related Λcoalescents. If T (n) denotes the length of an external branch of the ncoalescent, we prove the convergence of n α−1 T (n) when n tends to ∞, and give the limit. To this aim, we give asymptotics for the number σ (n) of collisions which occur in the ncoalescent until the end of the chosen external branch, and for the block counting process associated with the ncoalescent.