Random coverings in several dimensions
The author presents far-reaching generalizations of results on limit distributions for the number of random arcs covering a circle. He considers first a bounded set \(K\subset {\mathbb{R}}^ d\) and a sequence \(B_ 1,B_ 2,..\). of i.i.d. random convex sets which are uniformly distributed in a set \(V\subset {\mathbb{R}}^ d\) which contains the closure of K in its interior. The (minimal) number \(N_ m\) for which \(B_ 1,...,B_{N_ m}\) cover each point of K at least m times then has the following asymptotic distributional behaviour (as the size a of the random sets \(B_ i\) shrinks to zero): \[ (b(a)/| V|)N_ m-\log | K| /b(a)-(d+m-1)\cdot \log \log | K| /b(a)+\log (m- 1\quad)!-\log \alpha (K)\to U \] (in distribution). Here \(| \cdot |\) denotes the volume, b(a) is the expected volume of \(B_ 1\), and U has the extreme value distribution \(P(U\leq u)=\exp (-\exp (-u))\). A similar result is shown for a compact Riemannian manifold which is covered by random geodesic balls. Finally, it is proved (under a weak condition on the boundary of K) that the number \(N_ m'\) of sets in \(B_ 1,...,B_{N_ m}\) which actually meet K has the same limit behaviour as \(N_ m\) with \(| V|\) replaced by \(| K|\). For the proofs some interesting results on intersections of convex sets are established. Also, the constant \(\alpha\) (K) which is the only term depending on the shape of K is discussed and some open problems are posed.
scientific article; zbMATH DE number 3962850