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

Understand

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.