On the coverage of k-dimensional space by k-dimensional spheres. (English) Zbl 0582.60015

Hall, Peter

Ann. Probab. 13, 991-1002 (1985).

The asymptotic properties of coverage probabilities are investigated for n balls each of content \(a_ n\) distributed independently within a unit k-dimensional cube with toroidal edge correction. In the case of uniformly distributed balls, a necessary and sufficient condition for the probability of the entire cube being covered at least \(\ell\) times to tend to 1 as \(n\to \infty\) is that \(na_ n-\log n-(k+\ell -2) \log \log n\to \infty.\) An analogous result holds for spheres located according to a homogeneous Poisson point process as the intensity parameter tends to \(\infty.\)
In the case of a non-uniform density f governing the sphere centres, the corresponding condition is \(na_ n f(m)-\log n-(k/2+\ell -2) \log \log n\to \infty.\) Here the unique, non-zero minimum of f occurs at m, and it is assumed that f is bounded with continuous, positive definite second order derivatives at m. In the uniform case, lower and upper bounds of the same order are provided for the probability of complete \(\ell\)- coverage.

Reviewer: P.Davy

MSC:

60D05	Geometric probability and stochastic geometry
60E05	Probability distributions: general theory
52C17	Packing and covering in \(n\) dimensions (aspects of discrete geometry)

Keywords:

asymptotic probability; coverage probabilities; Poisson point process