  For geometrically finite group actions on hyperbolic metric spaces and under
certain assumptions on the growth of parabolic subgroups, we prove a global
shadow lemma for Patterson-Sullivan measures, as well as a Dirichlet-type
theorem and a logarithm law for excursion of geodesics into cusps. We then
apply these results to geometrically finite quotients of strictly convex
Hilbert geometries with C^1 boundary.
