Abstract:
Variants of the minimal Lp-metrics are introduced in the
space of random measures and are applied, with various modifications of the
contraction method, to prove existence and uniqueness results for self-similar
random fractal measures. We obtain exponential convergence, both in distribution
and almost surely, of an iterative sequence of random measures (defined by
means of the scaling operator)
to a unique self-similar random measure. The assumptions are quite weak, and
correspond to similar conditions in the deterministic case.
The fixed mass case is handled in a direct way based on regularity
properties of the metrics and the properties of a natural
probability space. To prove convergence in the random mass case
needs additional tools such as a specially adapted choice of the
space of random measures and of the space of probability
distributions on measures, the introduction of reweighted sequences
of random measures and a comparison technique.
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