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

## Abstract

Let ξ_{1}, ξ_{2},… be a Poisson point process of density λ on (0,∞)^{d}, d ≥ 1, and let ρ, ρ_{1}, ρ_{2},… 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 X_{1}, X_{2},… is a {0,1}-valued Markov chain and ρ_{1}, ρ_{2},… 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 |
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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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