  Using techniques from ergodic theory and symbolic dynamics, we derive
statistical limit laws for real valued functions on hyperbolic groups. In
particular, our results apply to convex cocompact group actions on
$\text{CAT}(-1)$ spaces, and provide a precise statistical comparison between
word length and displacement. After generalising our methods to the
multidimensional setting, we prove that the abelianisation map satisfies a
non-degenerate multidimensional central limit theorem. We also obtain local
limit theorems for group homomorphisms and for the displacement function
associated to certain actions.
