On Asymptotics of Means of Non-Euclidean Data
Stephan Huckemann (Goettingen)
In many applications, data occur on non-Euclidean spaces. Simple examples are wind
directions on the circle and geological crack directions on the sphere. More advanced ones
are shapes of geometrical objects and phylogenetic trees which lead to so called stratified
spaces. Since all of these spaces are in particular metric spaces, means and expected
elements with respect to a squared distance can be defined. How to choose from a multitude
of canonical distances, however, is often not clear. While linking the central limit
theorem for large sample statistics to specific distances we find desirable properties,
such as "manifold stability" keeping expected elements away from singularities on
non-manifold G-spaces, less desirable properties, such that the distributional behavior at
the cut locus may govern the rate of convergence, and undesirable properties such as
stickiness" forcing sample means to hit singularities in finite time on non-manifold CAT(0)
spaces.