Abstract
In this paper we deal with the classical problem of random cover times. We investigate the distribution of the time it takes for a Poisson process of cylinders to cover a set A subset of R-d . This Poisson process of cylinders is invariant under rotations, reflections and translations, and in addition we add a time component so that cylinders are raining from the sky at unit rate. Our main results concerns the asymptotic of this cover time as the set A grows. If the set A is discrete and well separated, we show convergence of the cover time to a Gumbel distribution. If instead A has positive box dimension (and satisfies a weak additional assumption), we find the correct rate of convergence.
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