Discrete One-Dimensional Coverage Process on a Renewal Process

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.