WSEAS Transactions on Systems
Print ISSN: 1109-2777, E-ISSN: 2224-2678
Volume 17, 2018
Random Cutouts of the d-Dimensional Balls with i.i.d. Centers
Authors: ,
Abstract: Consider the random open balls $$B_{n}(ω)$$ with their centers $$ω_{n}$$ independently and uniformly distributed
over the d-dimensional unit cube $$[0, 1]^d$$ and with their radii $$r_{n}$$ decreasing to zero. In this paper, we show that with
probability one Hausdorff dimension of the random cut-out set $$[0, 1]^d-U_{n-1}^\infty B_{n}(ω)$$ is at most $$d-\frac{β(d)c^d}{p}$$ frequently equals $$d-\frac{β(d)c^d}{p}$$ when $$r_{n}=\frac{c}{n^p}$$ for some $$0< c < \sqrt[d]{β(d)} $$ and $$pd = 1$$