Random iteration of isometries in unbounded metric spaces

We consider an iterated function system (with probabilities) of isometrics on an unbounded metric space (X, d). Under suitable conditions it is proved that the random orbit {Z(n)}(ngreater than or equal to0) of the iterations corresponding to an initial point Z(0) is an element of X 'escapes to infinity' in the sense that P(Z(n) is an element of K) --> 0, as n --> infinity for every bounded set K subset of X. As an application we prove the corresponding result in the Euclidean and hyperbolic spaces under the condition that the isometries do not have a common fixed point.