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On the coverage of space by random sets

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Siva Athreya ,
Rahul Roy  and
Anish Sarkar
Siva Athreya*
Affiliation:
Indian Statistical Institute, Delhi
Rahul Roy*
Affiliation:
Indian Statistical Institute, Delhi
Anish Sarkar*
Affiliation:
Indian Statistical Institute, Delhi
*
∗ Postal address: Indian Statistical Institute, 7 S.J.S. Sansanwal Marg, New Delhi 110016, India.
∗ Postal address: Indian Statistical Institute, 7 S.J.S. Sansanwal Marg, New Delhi 110016, India.
∗ Postal address: Indian Statistical Institute, 7 S.J.S. Sansanwal Marg, New Delhi 110016, India.
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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 {ξi + [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.

Keywords

Complete coveragerenewal theoremMarkov chainPoisson processBoolean model

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 36 , Issue 1 , March 2004 , pp. 1 - 18
Copyright
Copyright © Applied Probability Trust 2004 

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