The internal branch lengths of the Kingman coalescent

Abstract

In the Kingman coalescent tree the length of order $r$ is defined as the sum of the lengths of all branches that support $r$ leaves. For $r=1$ these branches are external, while for $r\ge 2$ they are internal and carry a subtree with $r$ leaves. In this paper we prove that for any $s\in \mathbb{N}$ the vector of rescaled lengths of orders $1\le r\le s$ converges to the multivariate standard normal distribution as the number of leaves of the Kingman coalescent tends to infinity. To this end we use a coupling argument which shows that for any $r\ge 2$ the (internal) length of order $r$ behaves asymptotically in the same way as the length of order 1 (i.e., the external length).

Published at http://dx.doi.org/10.1214/14-AAP1024 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Beta-coalescents and continuous stable random trees
• Asympotic behavior of the total length of external branches for Beta-coalescents

Cited by in corpus (1)

• A compendium of covariances and correlation coefficients of coalescent tree properties