  We prove a formula for the speed of distance stationary random sequences. A
particular case is the classical formula for the largest Lyapunov exponent of
an i.i.d. product of two by two matrices in terms of a stationary measure on
projective space. We apply this result to Poisson-Delaunay random walks on
Riemannian symmetric spaces. In particular, we obtain sharp estimates for the
asymptotic behavior of the speed of hyperbolic Poisson-Delaunay random walks
when the intensity of the Poisson point process goes to zero. This allows us to
prove that a dimension drop phenomena occurs for the harmonic measure
associated to these random walks. With the same technique we give examples of
co-compact Fuchsian groups for which the harmonic measure of the simple random
walk has dimension less than one.
