We prove asymptotic normality of the total length of external branches in the Kingman coalescent. The proof uses an embedded Markov chain, which can be described as follows: Take an urn with black balls. Empty it step by step 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 of red balls form a Markov chain with an unexpected property: It is time-reversible.
"On the Total External Length of the Kingman Coalescent." Electron. J. Probab. 16 2203 - 2218, 2011. https://doi.org/10.1214/EJP.v16-955