  We establish Poisson and compound Poisson approximations for stabilizing
statistics of $\beta$-mixing point processes and give explicit rates of
convergence. Our findings are based on a general estimate of the total
variation distance of a stationary $\beta$-mixing process and its Palm version.
As main contributions, this article (i) extends recent results on Poisson
process approximation to non-Poisson/binomial input, (ii) gives concrete bounds
for compound Poisson process approximation in a Wasserstein distance and (iii)
illustrates the applicability of the general result in an example on minimal
angles in the stationary Poisson-Delaunay tessellation. The latter is among the
first (nontrivial) situations in Stochastic Geometry, where compound Poisson
approximation can be established with explicit extremal index and cluster size
distribution.
