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ASYMPOTIC BEHAVIOR OF THE TOTAL LENGTH OF EXTERNAL BRANCHES FOR BETACOALESCENTS
"... Abstract. In this paper, we consider the Beta(2 − α,α)coalescents with 1 < α < 2 and study the moments of external branches, in particular the total external branch length L (n) ext of an initial sample of n individuals. For this class of coalescents, it has been proved that n α−1 (n) (d) T → ..."
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Abstract. In this paper, we consider the Beta(2 − α,α)coalescents with 1 < α < 2 and study the moments of external branches, in particular the total external branch length L (n) ext of an initial sample of n individuals. For this class of coalescents, it has been proved that n α−1 (n) (d) T → T, where T (n) is the length of an external branch chosen at random, and T is a known non negative random variable. We get the asymptotic behaviour of several moments of L (n) ext. As a consequence, we obtain that for Beta(2−α,α)coalescents with 1 < α < 2, lim n→+ ∞ n3α−5 E[(L (n) ext −n2−α E[T]) 2] =
Asymptotic sampling formulae for Λcoalescents
, 2012
"... We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a Λcoalescent. This allows us to derive an exa ..."
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We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a Λcoalescent. This allows us to derive an exact formula for the asymptotic behavior of the site and allele frequency spectrum and the number of segregating sites, as the sample size tends to ∞. Some of our results hold in the case of a general Λcoalescent that comes down from infinity, but we obtain more precise information under a regular variation assumption. In this case, we obtain results of independent interest for the time at which a mutation uniformly chosen at random was generated. This exhibits a phase transition at α = 3/2, where α ∈ (1, 2) is the exponent of regular variation. AMS 2000 Subject Classification. 60J25, 60F99, 92D25 Key words and phrases. Λcoalescents, speed of coming down from infinity, exchangeable
4 The Kingman tree length process has infinite quadratic variation∗
, 2014
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Asymptotics of the minimal clade size and related functionals of certain betacoalescents
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PARETO GENEALOGIES ARISING FROM A POISSON BRANCHING EVOLUTION MODEL WITH SELECTION
, 2013
"... Abstract. We study a class of coalescents derived from a sampling procedure out of N i.i.d. Pareto(α) random variables, normalized by their sum, including β−sizebiasing on total length effects (β < α). Depending on the range of α, we derive the large N limit coalescents structure, leading either ..."
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Abstract. We study a class of coalescents derived from a sampling procedure out of N i.i.d. Pareto(α) random variables, normalized by their sum, including β−sizebiasing on total length effects (β < α). Depending on the range of α, we derive the large N limit coalescents structure, leading either to a discretetime PoissonDirichlet(α,−β) Ξ−coalescent (α ∈ [0,1)), or to a family of continuoustime Beta(2−α,α−β) Λ−coalescents (α ∈ [1,2)), or to the Kingman coalescent (α ≥ 2). We indicate that thisclass ofcoalescent processes (and their scaling limits) may be viewed as the genealogical processes of some forward in time evolving branching population models including selection effects. In such constantsize population models, the reproduction step, which is based on a fitnessdependent Poisson Point Process with scaling powerlaw(α) intensity, is coupled to a selection step consisting of sorting out the N fittest individuals issued from the reproduction step. Running title: Pareto genealogies in a Poisson evolution model with selection.