Central limit theorems for Green metrics on hyperbolic groups

Suppose we have two finitely supported, admissible, probability measures on a hyperbolic group $\mathrm{\Gamma }$. In this article we prove that the corresponding two Green metrics satisfy a counting central limit theorem when we order the elements of $\mathrm{\Gamma }$ according to one of the metrics. Our results also apply to various other metrics including length functions associated to Anosov representations and to group actions on hyperbolic metric spaces.