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Random fractals and probability metrics

Part of: Stochastic processes Classical measure theory Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

John E. Hutchinson  and
Ludger Rüschendorf
John E. Hutchinson*
Affiliation:
Australian National University
Ludger Rüschendorf*
Affiliation:
Universität Freiburg
*
∗ Postal address: School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia.
∗∗ Postal address: Institut für Mathematische Stochastik, Universität Freiburg, Eckerstr. 1, D-79104 Freiburg, Germany. Email address: ruschen@stochastik.uni-freiburg.de
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Abstract

New 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. Proving 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.

Keywords

Random measurefractalself-similarprobability metricrandom fractalMonge-Kantorovich metricscaling operatoriterated function systemminimal metric

MSC classification

Primary: 60G57: Random measures
Secondary: 28A80: Fractals 60D05: Geometric probability and stochastic geometry 60G18: Self-similar processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 4 , December 2000 , pp. 925 - 947
Copyright
Copyright © Applied Probability Trust 2000 

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