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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 45 (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.
Asymptotic results on the length of coalescent trees
 Ann. Appl. Prob
"... Abstract. We give the asymptotic distribution of the length of partial coalescent trees for Beta and related coalescents. This allows us to give the asymptotic distribution of the number of (neutral) mutations in the partial tree. This is a first step to study the asymptotic distribution of a natura ..."
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Cited by 22 (2 self)
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Abstract. We give the asymptotic distribution of the length of partial coalescent trees for Beta and related coalescents. This allows us to give the asymptotic distribution of the number of (neutral) mutations in the partial tree. This is a first step to study the asymptotic distribution of a natural estimator of DNA mutation rate for species with large families. 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.
On asympotics of the betacoalescents
 arXiv:1203.3110, 2012. 20 ROMAIN ABRAHAM AND JEANFRANÇOIS
"... We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1, b) measure converges in distribution to a 1stable law, as the initial number of particles goes to infinity. The stable limit law is also shown for the total branch length of the coalescent tree ..."
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We show that the total number of collisions in the exchangeable coalescent process driven by the beta (1, b) measure converges in distribution to a 1stable law, as the initial number of particles goes to infinity. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance b = 1, which corresponds to the Bolthausen–Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta (a, b)coalescents with 0 < a < 1 leads to a simplified derivation of the known (2 − a)stable limit. We furthermore derive asymptotic expansions for the (centered) moments of the number of collisions and of the total branch length for the beta (1, b)coalescent by exploiting the method of sequential approximations. 1
The number of small blocks in exchangeable random partitions
, 2010
"... Suppose Π is an exchangeable random partition of the positive integers and Πn is its restriction to {1,...,n}. Let Kn denote the number of blocks of Πn, and let Kn,r denote the number of blocks of Πn containing r integers. We show that if 0 < α < 1 and Kn/(n α ℓ(n)) converges in probability t ..."
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Cited by 4 (0 self)
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Suppose Π is an exchangeable random partition of the positive integers and Πn is its restriction to {1,...,n}. Let Kn denote the number of blocks of Πn, and let Kn,r denote the number of blocks of Πn containing r integers. We show that if 0 < α < 1 and Kn/(n α ℓ(n)) converges in probability to Γ(1 − α), where ℓ is a slowly varying function, then Kn,r/(n α ℓ(n)) converges in probability to αΓ(r − α)/r!. This result was previously known when the convergence of Kn/(n α ℓ(n)) holds almost surely, but the result under the hypothesis of convergence in probability has significant implications for coalescent theory. We also show that a related conjecture for the case when Kn grows only slightly slower than n fails to be true.