Convergence of random walks to Brownian motion in phylogenetic tree-space

The set of all phylogenetic trees for a fixed set of species forms a geodesic metric space known as Billera-Holmes-Vogtmann tree-space. In order to analyse samples of phylogenetic trees it is desirable to construct parametric distributions on this space, but this task is very challenging. One way to construct such distributions is to consider particles undergoing Brownian motion in tree-space from a fixed starting point. The distribution of the particles after a given duration of time is analogous to a multivariate normal distribution in Euclidean space. Since these distributions cannot be worked with directly, we consider approximating by suitably defined random walks in tree-space. We prove that as the number of steps tends to infinity and the step-size tends to zero, the distributions obtained by random walk converge to those corresponding to Brownian motion. This result opens the possibility of statistical modelling using distributions obtained from Brownian motion on tree-space.