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Longest visible rays in Boolean models with general convex grains

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  • Thäle, Christoph

Abstract

We consider a stationary Boolean model in Rd whose grains are random compact convex bodies with non-empty interior. Conditionally on the origin being uncovered, we study the logarithmic tail asymptotics of the length Lo of the longest visible ray starting at the origin. For a fixed direction u∈Sd−1, the visible range has an exponential tail with rate equal to the intensity times the mean (d−1)-dimensional volume of the projection of the typical grain onto u⊥. Optimizing over all directions therefore leads to an anisotropic variational principle. We show that the decay rate for Lo is determined by the minimum mean brightness of the typical grain.

Suggested Citation

  • Thäle, Christoph, 2026. "Longest visible rays in Boolean models with general convex grains," Statistics & Probability Letters, Elsevier, vol. 238(C).
    • Handle: RePEc:eee:stapro:v:238:y:2026:i:c:s0167715226002439
    DOI: 10.1016/j.spl.2026.110879
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