# Discrete One-Dimensional Coverage Process on a Renewal Process

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## Abstract

Consider the following coverage model on \(\mathbb {N}\), for each site \(i \in \mathbb {N}\) associate a pair \((\xi _i, R_i)\) where \((\xi _i)_{i \ge 0}\) is a 1-dimensional undelayed discrete renewal point process and \((R_i)_{i \ge 0}\) is an i.i.d. sequence of \(\mathbb {N}\)-valued random variables. At each site where \(\xi _i=1\) start an interval of length \(R_i\). Coverage occurs if every site of \(\mathbb {N}\) is covered by some interval. We obtain sharp conditions for both, positive and null probability of coverage. As corollaries, we extend results of the literature of rumor processes and discrete one-dimensional Boolean percolation.

## Keywords

Coverage process Rumor process Renewal theory## Notes

### Acknowledgements

Sandro Gallo thanks FAPESP 2015/09094-3 and research fellowships CNPq (312315/2015-5) and FAPESP (2017/07084-6). Nancy Garcia thanks FAPESP 2014/26419-0 and research fellowship CNPq 302598/2014-6. This article was produced as part of the activities of FAPESP Research, Innovation and Dissemination Center for Neuromathematics (Grant #2013/07699-0 , São Paulo Research Foundation) and Edital Universal CNPq (480108/2012-9 and 462064/2014-0).

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