The external lengths in Kingman's coalescent

Svante Janson, Götz Kersting

Published 2010-04-28, updated 2011-01-17Version 2

In this paper we prove asymptotic normality of the total length of external branches in Kingman's coalescent. The proof uses an embedded Markov chain, which can be descriped as follows: Take an urn with n black balls. Empty it in n steps according to the rule: In each step remove a randomly chosen pair of balls and replace it by one red ball. Finally remove the last remaining ball. Then the numbers U_k, 0 \leq k \leq n, of red balls after k steps exhibits an unexpected property: (U_0,...,U_n) and (U_n,..., U_0) are equal in distribution.