Summary
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.
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Dawson, D.A., Hochberg, K.J.: The carrying dimension of a stochastic measure diffusion, Ann. Probab. 7, 693–703 (1979)
Falconer, K.J.: Random fractals. Math. Proc. Camb. Phil. Soc. 100, 559–582 (1986)
Federer, H.: Geometric measure theory, Berlin Heidelberg New York: Springer 1969
Graf, S.: Statistically self-similar fractals, Prob. Th. Rel. Fields 74, 357–392 (1987)
Graf, S., Mauldin, R.D., Williams, S.C.: The exact Hausdorff dimension in random recursive constructions, Memoirs AMS 71, 381 (1988)
Hutchinson, J.E.: Fractals and self similarity. Indiana Univ. Math. J. 30, 713–747 (1981)
Kallenberg, O.: Random measures. Berlin: Akademie-Verlag 1983
Kerstan, J., Matthes, K., Mecke, J.: Infinitely divisible point processes, Berlin: Akademie-Verlag 1974 Chichester-New York-Brisbane-Toronto: Wiley 1978/Moscou: Nauka 1982
Knight, F.B.: Essentials of Brownian motion and diffusion. Providence: AMS 1981
Mandelbrot, B.: Sporadic random functions and conditional sprectal analysis: self-similar examples and limits. Proceedings, Fifth Berkely Symposium on Math. Stat. and Probab., vol. III, pp. 155–179. Berkely: University of California Press 1967
Mandelbrot, B.: Renewal sets and random cutouts. Z. Wahrscheinlichkeitstheor. Verw. Geb. 22, 145–157 (1972)
Mandelbrot, B.: The fractal geometry of nature. San Francisco: Freeman 1983
Mauldin, R.D., Williams, S.C.: Random recursive constructions: asymptotic geometric and topological properties. Trans. Am. Math. Soc. 295, 325–346 (1986)
Smit, J.C.: Solution to problem 130. Statist. Neerlandica 37, 87 (1983)
Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic geometry and its applications, Berlin: Akademie-Verlag/Chichester-New York-Brisbane-Toronto-Singapore: Wiley 1987
Verwaat, W.: Sample path properties of self-similar processes with stationary increments, Ann. Probab. 13, 1–27 (1985)
Zähle, U.: Random fractals generated by random cutouts, Math. Nachr. 116, 27–52 (1984)
Zähle, U.: The fractal character of localizable measure-valued processes. III. Fractal carrying sets of branching diffusions, Math. Nachr. 138, (1988)
Zähle, U.: Self-similar random measures. II — A generalization: self-affine measures. Math. Nachr. (to appear)
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Zähle, U. Self-similar random measures. Probab. Th. Rel. Fields 80, 79–100 (1988). https://doi.org/10.1007/BF00348753
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DOI: https://doi.org/10.1007/BF00348753