Article,
Random Fractals Generated by Random Cutouts
Mathematische Nachrichten, 116 (1): 27--52 (1984)
DOI: 10.1002/mana.19841160104
Abstract
MR: The author is concerned with the fractal character of random sets in RN, or more generally, in a locally compact second-countable Hausdorff space. He generalizes a result due to B. B. Mandelbrot , who discussed the case N=1 and segmental cutout sets. A purely measure-geometric approach is used here. Let Ξ be a random closed set in RN. A constant D is called the essential dimension of Ξ if P(dimΞ≤D)=1 and P(dimΞ≥D)>0, where dimΞ is the Hausdorff-Besicovitch dimension of Ξ. In the first section a lower bound for dimΞ is constructed for general Ξ. In Section 2, Ξ is specified to be the limit of a decreasing sequence Ξn and a lower bound for dimΞ is given under conditions on the probability P(x1∈Ξn,x2∈Ξn,x3∈Ξn). In Section 3, a further specification of the form Ξn=RN∖⋃nk=1Γk is proposed, where the Γk's are independent and third-order stationary random open sets, that is, Ξ is generated by cutouts of the Γk's. For any Borel set B with positive Lebesgue measure and an increasing sequence rn, a condition for
D=N−limsupln(1/P(x∈Ξn))lnrn
to be a lower bound for essdim(Ξ∩B) is given and, in Section 4, also a condition for D to be an upper bound for essdim(Ξ∩B) is given. Simple sufficient conditions for the validity of the above two estimates, hence sufficient conditions for essdim(Ξ∩B)=D, are considered in Section 5. In Section 6, the case when the Γk's are similar, i.e., the distributions of rkΓk are identical, and, in Section 7, the case when the Γk's are generalized Boolean models, are discussed. In Section 8, the preceding results are applied to the covering problem, that is, the problem whether or not Ξ=∅. In the final section, essential dimensions of lower-dimensional sections of Ξ are investigated.
