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We study the asymptotic nature of geometric structures formed from a pointcloud of observations of (generally heavy tailed) distributions in a Euclideanspace of dimension greater than one. A typical example is given by the Bettinumbers of \v{C}ech complexes built over the cloud. The structure of dependenceand sparcity (away from the origin) generated by these distributions leads tolimit laws expressible via non-homogeneous, random, Poisson measures. Theparametrisation of the limits depends on both the tail decay rate of theobservations and the particular geometric constraint being considered.  The main theorems of the paper generate a new class of results in the wellestablished theory of extreme values, while their applications are ofsignificance for the fledgling area of rigorous results in topological dataanalysis. In particular, they provide a broad theory for the empiricallywell-known phenomenon of homological `crackle'; the continued presence ofspurious homology in samples of topological structures, despite increasedsample size.

Limit Theorems for Point Processes under Geometric Constraints (and Topological Crackle)