  We establish central limit theorems for an action of a group G on a
hyperbolic space X with respect to the counting measure on a Cayley graph of G.
Our techniques allow us to remove the usual assumptions of properness and
smoothness of the space, or cocompactness of the action. We provide several
applications which require our general framework, including to lengths of
geodesics in geometrically finite manifolds and to intersection numbers with
submanifolds.
