Multivariate statistics takes great advantage from linearization due to the linear structure of the data space. If data reside in non-linear spaces, which, for instance, occurs in shape analysis, already the most basic statistical concepts such as means or principal components can no longer be simply defined as arithmetic averages or eigenspaces of covariance matrices.

For suitable data descriptors living on manifolds or stratified spaces we investigate asymptotic properties to allow for statistical inference. It turns out that rates and central limit theorems may reflect the topological and geometric structure of the underlying space.