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On the coverage of space by random sets

  • Siva Athreya (a1), Rahul Roy (a1) and Anish Sarkar (a1)

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 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:X i =1} [i,i i ], where X 1, X 2,… 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.

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Corresponding author

Postal address: Indian Statistical Institute, 7 S.J.S. Sansanwal Marg, New Delhi 110016, India.
∗∗ Email address: athreya@isid.ac.in

References

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Keywords

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On the coverage of space by random sets

  • Siva Athreya (a1), Rahul Roy (a1) and Anish Sarkar (a1)

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