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We prove a formula for the speed of distance stationary random sequences. Aparticular case is the classical formula for the largest Lyapunov exponent ofan i.i.d. product of two by two matrices in terms of a stationary measure onprojective space. We apply this result to Poisson-Delaunay random walks onRiemannian symmetric spaces. In particular, we obtain sharp estimates for theasymptotic behavior of the speed of hyperbolic Poisson-Delaunay random walkswhen the intensity of the Poisson point process goes to zero. This allows us toprove that a dimension drop phenomena occurs for the harmonic measureassociated to these random walks. With the same technique we give examples ofco-compact Fuchsian groups for which the harmonic measure of the simple randomwalk has dimension less than one.

A Furstenberg type formula for the speed of distance stationary sequences