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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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References

[1] Ewens, W. J. and Grant, G. R. (2001). Statistical Methods in Bioinformatics: An Introduction. Springer, Berlin.Google Scholar
[2] Feller, W. (1968). Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
[3] Hall, P. (1988). Introduction to the Theory of Coverage Processes. John Wiley, New York.Google Scholar
[4] Knopp, K. (1949). Theory and Application of Infinite Series. Blackie, Glasgow.Google Scholar
[5] Mandelbrot, B. (1972). On the Dvoretzky coverings for the circle. Z. Wahrscheinlichkeitsth. 22, 158–160.CrossRefGoogle Scholar
[6] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge University Press.Google Scholar
[7] Molchanov, I. and Scherbakov, V. (2003). Coverage of the whole space. Adv. Appl. Prob. 35, 898–912.CrossRefGoogle Scholar
[8] Shepp, L. (1972). Covering the circle with random arcs. Israel J. Math. 11, 328–345.Google Scholar
[9] Tanemura, H. (1993). Behaviour of the supercritical phase of a continuum percolation model on Rd . J. Appl. Prob. 30, 382–396.CrossRefGoogle Scholar