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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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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.
Interpreting Λcoalescent speed of coming down from infinity via particle representation of superprocesses
 In preparation
, 2008
"... Consider a Λcoalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number Nt of blocks at any positive time t> 0). We exhibit a deterministic function v: (0,∞) → (0,∞), such that Nt/v(t) → 1, almost ..."
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Cited by 29 (9 self)
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Consider a Λcoalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number Nt of blocks at any positive time t> 0). We exhibit a deterministic function v: (0,∞) → (0,∞), such that Nt/v(t) → 1, almost surely and in Lp for any p ≥ 1, as t → 0. Our approach relies on a novel martingale technique.
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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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.
The asymptotic distribution of the length of betacoalescent trees
 Ann. Appl. Probab
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3 How to use MIGRATE or why are Markov chain Monte Carlo programs difficult to use?
 POPULATION GENETICS FOR ANIMAL CONSERVATION
, 2009
"... Population genetic analyses often require the estimation of parameters such as population size and migration rates. In the 1960s, enzyme electrophoresis was developed; it was the first method to gather codominant data from many individuals in many populations relatively easily. Summary statistics m ..."
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Cited by 13 (0 self)
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Population genetic analyses often require the estimation of parameters such as population size and migration rates. In the 1960s, enzyme electrophoresis was developed; it was the first method to gather codominant data from many individuals in many populations relatively easily. Summary statistics methods, such as allelefrequency based Fstatistics (Wright 1951), were used to estimate population genetics parameters from these data sets. These methods matured and expanded into many variants that were enthusiastically accepted by many researchers. Fstatistics are still a hallmark of any population genetic study, especially in conservation genetics, although over the years, limitations have become evident (Neigel 2002). Many of these methods use restrictive assumptions, for example, disallowing mutation. Fstatistics, such as FST methods, are often employed on pairs of populations; this can lead to biased parameter estimates (cf. Beerli 2004; Slatkin 2005) and the reuse of data in these pairwise methods is undesirable from a statistical viewpoint. In 1982, Sir John Kingman developed the coalescence theory (Kingman 1982a, b). His overview of the developments of this theory (Kingman 2000) gives an interesting insight into the development of new ideas. This new development opened the door to methods in population genetics that go beyond the Fstatistics methods and have led to several theoretical breakthroughs (Hein et al. 2005; although inferences based on coalescence theory were not practicable until about 1995 because of computational constraints). In recent years, computerintensive programs that can estimate parameters using genetic data under various coalescent models have been developed; for example, programs that estimate gene...
In Bertorelle, Giorgio, Bruford, M W, Hauffe, Heidi C, Rizzoli, A, & Vernesi, C (Eds.), Population Genetics for Animal Conservation (pp. 4279). Cambridge University Press, Cambridge UK.
The total external branch length of beta coalescents. Preprint available on http://arxiv.org/abs/1212.6070
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ON THE EXTENDED MORAN MODEL AND ITS RELATION TO COALESCENTS WITH MULTIPLE COLLISIONS
, 2011
"... We study the asymptotics of the extended Moran model as the total population size N tends to infinity. Two convergence results are provided, the first result leading to discretetime limiting coalescent processes and the second result leading to continuoustime limiting coalescent processes. The lim ..."
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We study the asymptotics of the extended Moran model as the total population size N tends to infinity. Two convergence results are provided, the first result leading to discretetime limiting coalescent processes and the second result leading to continuoustime limiting coalescent processes. The limiting coalescent processes allow for multiple mergers of ancestral lineages (Λcoalescent). It is furthermore verified that any continuous time Λcoalescent (with Λ any probability distribution) can arise in the limit. Typical examples of extended Moran models are discussed, with an emphasis on models being in the domain of attraction of beta coalescents or Λcoalescents with Λ being log infinitely divisible.
Unifying vertical and nonvertical evolution: a stochasticARGbased framework
 Syst
, 2010
"... Abstract.—Evolutionary biologists have introduced numerous statistical approaches to explore nonvertical evolution, such as horizontal gene transfer, recombination, and genomic reassortment, through collections of Markovdependent gene trees. These tree collections allow for inference of nonvertical ..."
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Abstract.—Evolutionary biologists have introduced numerous statistical approaches to explore nonvertical evolution, such as horizontal gene transfer, recombination, and genomic reassortment, through collections of Markovdependent gene trees. These tree collections allow for inference of nonvertical evolution, but only indirectly, making findings difficult to interpret and models difficult to generalize. An alternative approach to explore nonvertical evolution relies on phylogenetic networks. These networks provide a framework to model nonvertical evolution but leave unanswered questions such as the statistical significance of specific nonvertical events. In this paper, we begin to correct the shortcomings of both approaches by introducing the “stochastic model for reassortment and transfer events ” (SMARTIE) drawing upon ancestral recombination graphs (ARGs). ARGs are directed graphs that allow for formal probabilistic inference on vertical speciation events and nonvertical evolutionary events. We apply SMARTIE to phylogenetic data. Because of this, we can typically infer a single most probable ARG, avoiding coarse population dynamic summary statistics. In addition, a focus on phylogenetic data suggests novel probability distributions on ARGs. To make inference with our model, we develop a reversible jump Markov chain Monte Carlo sampler to approximate the posterior distribution of SMARTIE. Using the BEAST phylogenetic
Analysis of DNA sequence variation within marine species using Beta–coalescents
 Theor. Popln Biol
, 2013
"... Abstract. We apply recently developed inference methods based on general coalescent processes to DNA sequence data obtained from various marine species. Several of these species are believed to exhibit socalled shallow gene genealogies, potentially due to extreme reproductive behaviour, e.g. via H ..."
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Abstract. We apply recently developed inference methods based on general coalescent processes to DNA sequence data obtained from various marine species. Several of these species are believed to exhibit socalled shallow gene genealogies, potentially due to extreme reproductive behaviour, e.g. via Hedgecock’s “reproduction sweepstakes”. Besides the data analysis, in particular the inference of mutation rates and the estimation of the (real) time to the most recent common ancestor, we briefly address the question whether the genealogies might be adequately described by socalled Beta coalescents (as opposed to Kingman’s coalescent), allowing multiple mergers of genealogies. The choice of the underlying coalescent model for the genealogy has drastic implications for the estimation of the above quantities, in particular the realtime embedding of the genealogy.