The Total External Branch Length of Beta-Coalescents†. (November 2014)

• Record Type: Journal Article
• Authors: DAHMER, IULIA; KERSTING, GÖTZ; WAKOLBINGER, ANTON; Broutin, Nicolas; Fill, James Allen; Nebel, Markus; Ward, Mark Daniel
• Is Part Of: Combinatorics, probability and computing. Volume 23:Number 6(2014:Nov.)
• Page Start: 1010
• Page End: 1027
• Publication Date: 2014-11
• Subjects: Combinatorial analysis -- Periodicals; Probabilities -- Periodicals; Computer science -- Mathematics -- Periodicals; 511.6
• DOI: 10.1017/S0963548314000297
• Languages: English
• ISSNs: 0963-5483

Abstract

<abstract abstract-type="normal"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>For 1 &lt; α &lt; 2 we derive the asymptotic distribution of the total length of <italic>external</italic> branches of a Beta(2 − α, α)-coalescent as the number <italic>n</italic> of leaves becomes large. It turns out that the fluctuations of the external branch length follow those of τ<italic>_n</italic>^2−α over the entire parameter regime, where τ<italic>_n</italic> denotes the random number of coalescences that bring the <italic>n</italic> lineages down to one. This is in contrast to the fluctuation behaviour of the total branch length, which exhibits a transition at <inline-formula><alternatives><inline-graphic xlink:href="ark:/27927/pgh1ffxdr4f" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math><![CDATA[$\alpha_0 = (1+\sqrt 5)/2$]]></tex-math></alternatives></inline-formula> ([18]).</p> </abstract>