The internal branch lengths of the Kingman coalescent

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.2 they are internal and carry a subtree with r leaves. In this paper we prove that for any s.N the vector of rescaled lengths of orders 1.r.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.2 the (internal) length of order r behaves asymptotically in the same way as the length of order 1 (i.e., the external length).