  We prove a formula for the speed of distance stationary random sequences
generalizing the law of large numbers of Karlsson and Ledrappier. A particular
case is the classical formula for the largest Lyapunov exponent of i.i.d.\
matrix products, but our result has applications in various different contexts.
In many situations it gives a method to estimate the speed, and in others it
allows to obtain results of dimension drop for escape measures related to
random walks. We show applications to stationary reversible random trees with
conductances, Bernoulli bond percolation of Cayley graphs, and random walks on
cocompact Fuchsian groups.
