On the coverage of space by random sets

Athreya, Siva ; Roy, Rahul ; Sarkar, Anish (2004) On the coverage of space by random sets Advances in Applied Probability, 36 (1). pp. 1-18. ISSN 0001-8678

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Official URL: http://projecteuclid.org/euclid.aap/1077134461

Related URL: http://dx.doi.org/10.1239/aap/1077134461


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≥1i + [0,ρi]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 Rd by such Poisson Boolean models. In addition, we consider the set {i≥1:Xi=1} [i,i+ρi], 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.

Item Type:Article
Source:Copyright of this article belongs to Applied Probability Trust.
Keywords:Complete Coverage; Renewal Theorem; Markov Chain; Poisson Process; Boolean Model
ID Code:72333
Deposited On:29 Nov 2011 13:39
Last Modified:29 Nov 2011 13:39

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