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Random fractals and probability metrics

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

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: [email protected]
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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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References

Arbeiter, M. A. (1991). Random recursive constructions of self-similar fractal measures. The noncompact case. Prob. Theor. Rel. Fields 88, 497–520.Google Scholar
Barnsley, M. F. (1988). Fractals Everywhere. Academic Press, New York.Google Scholar
Barnsley, M. F. and Demko, S. (1985). Iterated function systems and the global construction of fractals. Proc. R. Soc. London Ser. A 399, 243–275.Google Scholar
Barnsley, M. F. and Elton, J. H. (1985). Stationary attractive measures for a class of Markov chains arising from function iteration. Preprint.Google Scholar
Barnsley, M. F. and Elton, J. H. (1988). A new class of Markov processes for image encoding. Adv. Appl. Prob. 20, 14–32.CrossRefGoogle Scholar
Barnsley, M. F., Demko, S. G., Elton, J. H. and Geronimo, J. S. (1988). Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities. Ann. Inst. H. Poincaré Prob. Statist. 24, 367–394.Google Scholar
Barnsley, M. F., Demko, S. G., Elton, J. H. and Geronimo, J. S. (1989). Erratum: Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities. Ann. Inst. H. Poincaré Prob. Statist. 25, 589–590.Google Scholar
Falconer, K. J. (1986). Random fractals. Math. Proc. Camb. Phil. Soc. 100, 559–582.Google Scholar
Graf, S. (1987). Statistically self-similar fractals. Prob. Theor. Rel. Fields 74, 357–392.Google Scholar
Graf, S., Mauldin, R. D. and Williams, S. C. (1988). The exact Hausdorff dimension in random recursive constructions. Mem. Amer. Math. Soc. 71, No. 381.Google Scholar
Hutchinson, J. E. (1981). Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747.Google Scholar
Hutchinson, J. E. (2000). Deterministic and random fractals. In Complex Systems, eds Bossomaier, T. R. J. and Green, D. G., Cambridge University Press.Google Scholar
Hutchinson, J. E. and Rüschendorf, L. (1998). Random fractal measures via the contraction method. Indiana Univ. Math. J. 47, 471–487.Google Scholar
Hutchinson, J. E. and Rüschendorf, L. (2000). Selfsimilar fractals and selfsimilar random fractals. In Fractal Geometry and Stochastics II, eds Bandt, C., Graf, S. and Zähle, M.. Birkhäuser, Basel, pp. 109–123.Google Scholar
Kahane, J. P. and Peyrière, J. (1976). Sur certaines martingales de Mandelbrot. Adv. Math. 22, 131–145.Google Scholar
Mauldin, R. D. and Williams, S. C. (1986). Random recursive constructions; asymptotic geometric and topological properties. Trans. Amer. Math. Soc. 295, 325–346.Google Scholar
Müller, N., (1995). Zufällige und Multifraktale mit Anwendungen in der Computergrafik. , Univ. Freiburg.Google Scholar
Olsen, L. (1994). Random geometrically graph directed self-similar multifractals (Pitman Research Notes 307). Longman, Harlow.Google Scholar
Patzschke, N. and Zähle, U. (1990). Self-similar random measures. IV. The recursive construction model of Falconer, Graf, and Mauldin and Williams. Math. Nachr. 149, 285–302.Google Scholar
Peitgen, H. O. and Saupe, D. (1988). The Science of Fractal Images, Springer, New York.Google Scholar
Rachev, S. T. (1991). Probability Metrics and the Stability of Stochastic Models. John Wiley, New York.Google Scholar
Rachev, S. T. and Rüschendorf, L. (1995). Probability metrics and recursive algorithms. Adv. Appl. Prob. 27, 770–799.Google Scholar
Rösler, U., (1992). A fixed point theorem for distributions. Stoch. Proc. Appl. 42, 195–214.Google Scholar
Saupe, D. (1988). Algorithms for random fractals. In The Science of Fractal Images, eds Peitgen, H. O. and Saupe, D.. Springer, New York, pp. 71–113.Google Scholar
Zähle, U., (1988). Self-similar random measures. I. Notion, carrying Hausdorff dimension, and hyperbolic distribution. 80, 79–100.Google Scholar