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88
Convergence in distribution of random metric measure spaces (Λcoalescent measure trees)
, 2007
"... We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topol ..."
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Cited by 52 (10 self)
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We consider the space of complete and separable metric spaces which are equipped with a probability measure. A notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge. This topology is metrized following Gromov’s idea of embedding two metric spaces isometrically into a common metric space combined with the Prohorov metric between probability measures on a fixed metric space. We show that for this topology convergence in distribution follows provided the sequence is tight from convergence of all randomly sampled finite subspaces. We give a characterization of tightness based on quantities which are reasonably easy to calculate. Subspaces of particular interest are the space of real trees and of ultrametric spaces equipped with a probability measure. As an example we characterize convergence in distribution for the (ultra)metric measure spaces given by the random genealogies of the Λcoalescents. We show that the Λcoalescent defines an infinite (random) metric measure space if and only if the socalled “dustfree”property holds.
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.
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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Cited by 25 (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.
A modified lookdown construction for the XiFlemingViot process with mutation and populations with recurrent bottlenecks
 ALEA
, 2009
"... populations with recurrent bottlenecks ..."
A coalescent model for the effect of advantageous mutations on the genealogy of a population
, 2008
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A new model for evolution in a spatial continuum
"... o b a b i l i t y Vol. 15 (2010), Paper no. 7, pages 162–216. Journal URL ..."
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Cited by 20 (5 self)
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o b a b i l i t y Vol. 15 (2010), Paper no. 7, pages 162–216. Journal URL