Let G be a Gromov-hyperbolic group and S a finite symmetric generating set. The choice of S determines a metric on G (namely the graph metric on the associated Cayley graph). Given a representation of G in GL(d,R), we are interested in establishing statistical 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. 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.