Probability Theory and Related Fields

, Volume 97, Issue 4, pp 559–574

Self-similar random measures are locally scale invariant


  • N. Patzschke
    • Mathematische FakultätFriedrich-Schiller-Universität Jena
  • M. Zähle
    • Mathematische FakultätFriedrich-Schiller-Universität Jena

DOI: 10.1007/BF01192964

Cite this article as:
Patzschke, N. & Zähle, M. Probab. Th. Rel. Fields (1993) 97: 559. doi:10.1007/BF01192964


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.

Mathematics Subject Classification


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© Springer-Verlag 1993