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Self-similar random measures are locally scale invariant
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  • Published: December 1993

Self-similar random measures are locally scale invariant

  • N. Patzschke1 &
  • M. Zähle1 

Probability Theory and Related Fields volume 97, pages 559–574 (1993)Cite this article

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  • 10 Citations

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Summary

In an earlier paper Patzschke and U. Zähle [11] have proved the existence of a fractional tangent measure at the typical point of a self-similar random measure Φ under rather special technical assumptions. In the present paper we remove the most restrictive one. Here we suppose the open set condition for the similarities, a constant positive lower bound for the random contraction ratios, and vanishing Φ on the boundary of the open set with probability 1. The tangent measure isD-scale-invariant, whereD is the similarity dimension of Φ. Moreover, we approximate the tangential distribution by means of Φ and use this in order to prove that the Hausdorff dimension of the tangent measure equalsD. Since the former coincides with the Hausdorff dimension of Φ we obtain an earlier result of Mauldin and Williams [9] as a corollary.

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Authors and Affiliations

  1. Mathematische Fakultät, Friedrich-Schiller-Universität Jena, D-07740, Jena, Germany

    N. Patzschke & M. Zähle

Authors
  1. N. Patzschke
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  2. M. Zähle
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Patzschke, N., Zähle, M. Self-similar random measures are locally scale invariant. Probab. Th. Rel. Fields 97, 559–574 (1993). https://doi.org/10.1007/BF01192964

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  • Received: 27 March 1992

  • Revised: 31 May 1993

  • Issue Date: December 1993

  • DOI: https://doi.org/10.1007/BF01192964

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Mathematics Subject Classification

  • 60G57
  • 60D05
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