On the coverage of space by random sets

Abstract

Let ξ₁, ξ₂,… be a Poisson point process of density λ on (0,∞)^d, d ≥ 1, and let ρ, ρ₁, ρ₂,… be i.i.d. positive random variables independent of the point process. Let C := _i≥1 {ξ_i + [0,ρ_i]^d}. If, for some t > 0, (0,∞)^d C, then we say that (0,∞)^d is eventually covered by C. We show that the eventual coverage of (0,∞)^d depends on the behaviour of xP(ρ > x) as x → ∞ as well as on whether d = 1 or d ≥ 2. These results may be compared to those known for complete coverage of R^d by such Poisson Boolean models. In addition, we consider the set _{i≥1:Xi=1} [i,i+ρ_i], where X₁, X₂,… is a {0,1}-valued Markov chain and ρ₁, ρ₂,… are i.i.d. positive-integer-valued random variables independent of the Markov chain. We study the eventual coverage properties of this random set.