Central limit theorems for Fréchet means in the space of phylogenetic trees. (English) Zbl 1284.60023

Summary: This paper studies the characterisation and the limiting distributions of Fréchet means in the space of phylogenetic trees. This space is topologically stratified as well as being a CAT(0) space. We use a generalised version of the Delta method to demonstrate non-classical behaviour arising from the global topological structure of the space. In particular, we show that, for the space \(T_4\) of trees with four leaves, although they are related to the Gaussian distribution, the forms taken by the limiting distributions depend on the co-dimensions of the strata in which the Fréchet means lie.


60D05 Geometric probability and stochastic geometry
60F05 Central limit and other weak theorems
05C80 Random graphs (graph-theoretic aspects)
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