Probability Theory and Related Fields

, Volume 80, Issue 1, pp 79–100

Self-similar random measures

I. Notion, carrying hausdorff dimension, and hyberbolic distribution
  • U. Zähle

DOI: 10.1007/BF00348753

Cite this article as:
Zähle, U. Probab. Th. Rel. Fields (1988) 80: 79. doi:10.1007/BF00348753


A set is called self-similar if it is decomposable into parts which are similar to the whole. This notion was generalized to random sets. In the present paper an alternative, axiomatic approach is given which makes precise the following idea (using Palm distribution theory): A random set is statistically self-similar if it is statistically scale invariant with respect to any center chosen at random from that set. For these sets Hausdorff dimension coincides with an intrinsic self-similarity index.

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • U. Zähle
    • 1
  1. 1.Sektion MathematikFriedrich-Schiller-UniversitätJenaGerman Democratic Republic

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