  The (CLT) central limit theorems for generalized Frechet means (data
descriptors assuming values in stratified spaces, such as intrinsic means,
geodesics, etc.) on manifolds from the literature are only valid if a certain
empirical process of Hessians of the Frechet function converges suitably, as in
the proof of the prototypical BP-CLT (Bhattacharya and Patrangenaru (2005)).
This is not valid in many realistic scenarios and we provide for a new very
general CLT. In particular this includes scenarios where, in a suitable chart,
the sample mean fluctuates asymptotically at a scale $n^{\alpha}$ with
exponents ${\alpha} < 1/2$ with a non-normal distribution. As the BP-CLT yields
only fluctuations that are, rescaled with $n^{1/2}$ , asymptotically normal,
just as the classical CLT for random vectors, these lower rates, somewhat
loosely called smeariness, had to date been observed only on the circle (Hotz
and Huckemann (2015)). We make the concept of smeariness on manifolds precise,
give an example for two-smeariness on spheres of arbitrary dimension, and show
that smeariness, although "almost never" occurring, may have serious
statistical implications on a continuum of sample scenarios nearby. In fact,
this effect increases with dimension, striking in particular in high dimension
low sample size scenarios.
