Discrete One-Dimensional Coverage Process on a Renewal Process |
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Authors: | Sandro Gallo Nancy L Garcia |
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Institution: | 1.Departamento de Estatística,Universidade Federal de S?o Carlos,S?o Carlos,Brazil;2.Universidade Estadual de Campinas,Campinas,Brazil |
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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. |
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