Three limit theorems for vacancy in multivariate coverage problems
The three limit theorems of the title concern the asymptotic distribution of V, the volume left uncovered when a large number n of small spheres (each of volume \(a_ n)\) are placed at random (according to a fairly general distribution) in a unit cube in \({\mathbb{R}}^ k\). The first two theorems establish asymptotic normality of V when the sum of the sphere volumes \(na_ n\) either tends to a positive constant or tends to zero but not too fast. The third establishes a compound Poisson limit of a standardised version of V when \(n^ 2a_ n\) tends to zero. The method of proof for the first two theorems involves dissection of the cube into subcubes, moment estimates on the vacancy within slightly shrunk subcubes, and the Berry-Esseen theorem. For the third theorem estimates are made of the number of intersecting spheres.
scientific article; zbMATH DE number 3971867