Let $\Gamma$ be a Gromov-hyperbolic group and $S$ a finite symmetric generating set. The choice of $S$ determines a metric on $\Gamma$ (namely the graph metric on the associated Cayley graph). Given a representation $\rho: \Gamma \to GL_d(\mathbb R)$, we are interested in obtaining results analogous to random matrix products theory (RMPT) but for the deterministic sequence of spherical averages (with respect to $S$-metric). We will discuss a general law of large numbers and more refined limit theorems such as central limit theorem and large deviations. If time allows, we will also see boundary limit theorems and convergence of interpolated matrix norms along geodesic rays to the standard Brownian motion. The connections with (and results in) the classical RMPT, a result of Lubotzky–Mozes–Raghunathan and a question of Kaimanovich–Kapovich–Schupp will be discussed. Joint work with S. Cantrell.