Self-similar random measures

Summary A set is called self-similar if it is decomposable into parts which are similar to the whole. This notion was generalized to random sets. In the present paper an alternative, axiomatic approach is given which makes precise the following idea (using Palm distribution theory): A random set is statistically self-similar if it is statistically scale invariant with respect to any center chosen at random from that set. For these sets Hausdorff dimension coincides with an intrinsic self-similarity index.