  For sequences of random backward nested subspaces as occur, say, in dimension
reduction for manifold or stratified space valued data, asymptotic results are
derived. In fact, we formulate our results more generally for backward nested
families of descriptors (BNFD). Under rather general conditions, asymptotic
strong consistency holds. Under additional, still rather general hypotheses,
among them existence of a.s. local twice differentiable charts, asymptotic
joint normality of a BNFD can be shown. If charts factor suitably, this leads
to individual asymptotic normality for the last element, a principal nested
mean or a principal nested geodesic, say. It turns out that these results
pertain to principal nested spheres (PNS) and principal nested great subsphere
(PNGS) analysis by Jung et al. (2010) as well as to the intrinsic mean on a
first geodesic principal component (IMo1GPC) for manifolds and Kendall's shape
spaces. A nested bootstrap two-sample test is derived and illustrated with
simulations. In a study on real data, PNGS is applied to track early human
mesenchymal stem cell differentiation over a coarse time grid and, among
others, to locate a change point with direct consequences for the design of
further studies.
