  Recently, the first author and Tanaka obtained a large deviation principle
comparing two Green metrics on a non-elementary hyperbolic group $\Gamma$
associated to finitely supported, admissable probability measures. The aim of
this article is to prove the corresponding central limit theorem. Furthermore,
our results apply to various other metrics including length functions
associated to Anosov representations and to group actions on hyperbolic metric
spaces.
  In the particular case that $\Gamma$ is the fundamental group of a closed
hyperbolic manifold our work provides a statistical characterisation for the
hitting measure of a finitely supported random walk being in the same class as
the Lebesgue measure.
