Random Fractal Measures via the Contraction Method

In this paper we extend the contraction mapping method to prove various existence and uniqueness properties of (self-similar) random fractal measures, and establish exponential convergence results for approximating sequences defined by means of the scaling operator. For this purpose we introduce a version of the Monge Kantorovich metric on the class of probability distributions of random measures in order to prove the relevant results in distribution. We also use a special sample space of “construction trees” on which we define the approximating sequence of random measures, and introduce a certain operator and a compound variant of the Monge Kantorovich metric in order to establish a.s. exponential convergence to the unique random fractal measure. The arguments used apply at the random measure and random measure distribution levels, and the results cannot be obtained by previous contraction arguments which applied at the individual realisation level.