Brownian motion on the Yule tree: variance of the sample variance

Last modified: 2014-03-31

#### Abstract

We use the Yule tree conditioned on having n tips as a model of an unknown phylogenetic tree for n related species. The current trait values (X_1,...,X_n) are assumed to evolve from an ancestral state following the Brownian motion with the standard deviation parameter, describing the evolutionary rate, being equal 1. We consider the corresponding sample variance S^2 and demonstrate the following remarkable asymptotic result: the variance of S^2 tends to 1+pi^2/6 as n tends to infinity. This is a joint work with Krzysztof Bartoszek.

#### Keywords

Conditioned Yule process, Brownian motion, comparative phylogenetics