Discrete one-dimensional coverage process on a renewal process

2015/11/18 by Gallo, Sandro, Garcia, Nancy L.
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1511.05637

Abstract

We consider the following coverage model on ℕ. For each site i∈ ℕ we associate a pair (ξ_i, R_i) where \ξ₀, ξ₁, … \ is a 1-dimensional undelayed discrete renewal point process and \R₀,R₁,…\ is an i.i.d. sequence of ℕ-valued random variables. At each site where ξ_i=1 we start an interval of length R_i. Coverage occurs if every site of 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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