  We show that the displacement and translation distance of non-elementary
random walks on isometry groups of hyperbolic spaces satisfy large deviation
principles with the same rate function $I$. Roughly, this means that there
exists function $I(t)$ which accurately predicts the exponential decay rate of
the probability that the translation distance of a random product of length $n$
is $tn$, and similarly for the displacement. This settles a special case of a
conjecture concerning the large deviation principle for the spectral radius of
random matrix products. In a second part, we give a characterization of the
effective support of the rate function only in terms of the deterministic
notion of joint stable length. Finally, as a by-product of our techniques, we
deduce some further deterministic results on the asymptotics of a bounded set
of isometries. Some of the results in this paper were obtained simultaneously
and independently by Boulanger--Mathieu as we discuss in the introduction.
