Asympotic behavior of the total length of external branches for Beta-coalescents

2012/02/27 by Jean-Stephane Dhersin, Dhersin, Jean-Stephane, Linglong Yuan +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #FOS: Biological sciences #FOS: Mathematics #Populations and Evolution (q-bio.PE) #Probability (math.PR) #math.PR #q-bio.PE

paper · pdf · doi:10.48550/arxiv.1202.5859

arxiv created 2013/05/23 · arxiv updated 2013/05/24

Abstract

We consider a Λ-coalescent and we study the asymptotic behavior of the total length L⁽ⁿ⁾_ext of the external branches of the associated n-coalescent. For Kingman coalescent, i.e. Λ=δ₀, the result is well known and is useful, together with the total length L⁽ⁿ⁾, for Fu and Li's test of neutrality of mutations% under the infinite sites model asumption . For a large family of measures Λ, including Beta(2-α,α) with 0<α<1, Möhle has proved asymptotics of L⁽ⁿ⁾_ext. Here we consider the case when the measure Λ is Beta(2-α,α), with 1<α<2. We prove that n^α-2L⁽ⁿ⁾_ext converges in L² to α(α-1)Γ(α). As a consequence, we get that L⁽ⁿ⁾_ext/L⁽ⁿ⁾ converges in probability to 2-α. To prove the asymptotics of L⁽ⁿ⁾_ext, we use a recursive construction of the n-coalescent by adding individuals one by one. Asymptotics of the distribution of d normalized external branch lengths and a related moment result are also given.

Related

• On the length of an external branch in the Beta-coalescent
• Asymptotic results on the length of coalescent trees
• External branch lengths of Λ-coalescents without a dust component
• The external lengths in Kingman's coalescent
• The joint fluctuations of the lengths of the Beta(2-α, α)-coalescents