On the length of an external branch in the Beta-coalescent

2012/01/19 by Jean-Stephane Dhersin, Fabian Freund, Dhersin, Jean-Stephane +5
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.1201.3983

arxiv created 2012/01/19 · arxiv updated 2012/01/20

Abstract

In this paper, we consider Beta(2-α,α) (with 1<α<2) and related Λ-coalescents. If T⁽ⁿ⁾ denotes the length of an external branch of the n-coalescent, we prove the convergence of n^α-1T⁽ⁿ⁾ when n tends to ∞ , and give the limit. To this aim, we give asymptotics for the number σ⁽ⁿ⁾ of collisions which occur in the n-coalescent until the end of the chosen external branch, and for the block counting process associated with the n-coalescent.

Related

• The total external branch length of Beta-coalescents
• Asympotic behavior of the total length of external branches for Beta-coalescents
• The joint fluctuations of the lengths of the Beta(2-α, α)-coalescents
• Asymptotics of the minimal clade size and related functionals of certain beta-coalescents
• On the external branches of coalescents with multiple collisions