Limiting behaviour of Fréchet means in the space of phylogenetic trees (Q1695758)

The aim of the present paper is to study the limiting behaviour of the Fréchet means in the space \(T_m\) of phylogenetic trees with \(m\geq 5\) leaves that lie in top-dimensional or co-dimension one strata. In particular the authors managed to derive the limiting distributions for the corresponding sample Fréchet means generalizing their previous results on the space of phylogenetic trees with four leaves \(T_4\) regarding the important role of the topological structure of the space in the non-classical limiting behaviour of the Fréchet means in \(T_4\) (see [the authors, Electron. J. Probab. 18, Paper No. 25, 25 p. (2013; Zbl 1284.60023)]). Since the techniques used in those results cannot be adopted to analyze Fréchet means in the space \(T_m\), they managed at first to find the expression for the logmap of \(T_m\) using the geometric structure of geodesics in \(T_m\) proved by the third author [SIAM J. Discrete Math. 25, No. 4, 1506--1529 (2011; Zbl 1237.05045)], the third author and \textit{J. Provan} [``A fast algorithm for computing geodesic distances in tree space'', IEEE/ACM Trans. Comput. Biol. Bioinform. 8, No. 1, 2--13 (2011; \url{doi:10.1109/TCBB.2010.3})] and secondly to characterize the Fréchet means in \(T_m\) in terms of a modified version of this map finding out that although the Fréchet means limiting distributions are related to the Gaussian distribution they also depend on the co-dimensions of the strata in which the Fréchet means lie.