The external lengths in Kingman's coalescent

TL;DR

This paper proves that the total length of external branches in Kingman's coalescent follows an asymptotic normal distribution, using a novel Markov chain approach involving an urn model.

Contribution

It introduces a new Markov chain method to analyze the asymptotic distribution of external branch lengths in Kingman's coalescent.

Findings

External branch lengths are asymptotically normal.

The urn model reveals a symmetric distribution property.

Provides a new proof technique for coalescent properties.

Abstract

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.