  We consider the stochastic behavior of a class of local $U$-statistics of
Poisson processes$-$which include subgraph and simplex counts as special cases,
and amounts to quantifying clustering behavior$-$for point clouds lying in
diverging halfspaces. We provide limit theorems for distributions with light
and heavy tails. In particular, we prove finite-dimensional central limit
theorems. In the light tail case we investigate tails that decay at least as
slow as exponential and at least as fast as Gaussian. These results also
furnish as a corollary that $U$-statistics for halfspaces diverging at
different angles are asymptotically independent, and that there is no
asymptotic independence for heavy-tailed densities. Using state-of-the-art
bounds derived from recent breakthroughs combining Stein's method and Malliavin
calculus, we quantify the rate of this convergence in terms of Kolmogorov
distance. We also investigate the behavior of local $U$-statistics of a Poisson
Process conditioned to lie in diverging halfspace and show how the rate of
convergence in the Kolmogorov distance is faster the lighter the tail of the
density is.
