Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-27T20:59:03.380Z Has data issue: false hasContentIssue false

On the coverage of space by random sets

Published online by Cambridge University Press:  01 July 2016

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.

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≥1i + [0,ρi]d}. If, for some t > 0, (0,∞)dC, 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,ii], 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.

Type
Stochastic Geometry and Statistical Applications
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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, 158160.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, 898912.CrossRefGoogle Scholar
[8] Shepp, L. (1972). Covering the circle with random arcs. Israel J. Math. 11, 328345.Google Scholar
[9] Tanemura, H. (1993). Behaviour of the supercritical phase of a continuum percolation model on Rd . J. Appl. Prob. 30, 382396.CrossRefGoogle Scholar