  We study the asymptotic nature of geometric structures formed from a point
cloud of observations of (generally heavy tailed) distributions in a Euclidean
space of dimension greater than one. A typical example is given by the Betti
numbers of \v{C}ech complexes built over the cloud. The structure of dependence
and sparcity (away from the origin) generated by these distributions leads to
limit laws expressible via non-homogeneous, random, Poisson measures. The
parametrisation of the limits depends on both the tail decay rate of the
observations and the particular geometric constraint being considered.
  The main theorems of the paper generate a new class of results in the well
established theory of extreme values, while their applications are of
significance for the fledgling area of rigorous results in topological data
analysis. In particular, they provide a broad theory for the empirically
well-known phenomenon of homological `crackle'; the continued presence of
spurious homology in samples of topological structures, despite increased
sample size.
