Self-similar random measures

✍ Scribed by U. Zähle

• Publisher: Springer
• Year: 1988
• Tongue: English
• Weight: 939 KB
• Volume: 80
• Category: Article
• ISSN: 1432-2064
• DOI: 10.1007/bf00348753

✦ Synopsis

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.
The notion of self-similarity is of increasing interest in many theoretical and applied settings, such as random processes, turbulence, and dynamical systems, cf. e.g., Mandelbrot [-12J. Self-similar sets are usually fractal sets, i.e., sets of non-integer Hausdorff(-Besicovitch) dimension. Moreover, the parameters of selfsimilarity determine the Hausdorff dimension: If a compact set K is the union of sets K1, ...,KN, which are similar to K in the ratio r0 the "renormalized similitude" So,, in ~ by (So,~#)B=rD #(r-I B), t~eJg, B~ d.