Limit distribution of the sample volume fraction of Boolean set

We study the limit distribution of the volume fraction estimator p ̂ λ , A defined as the Lebesgue measure of the intersection X ∩ ( λ A ) of a random set X with a large observation set λ A , divided by the Lebesgue measure of λ A , as λ → ∞ , for a Boolean set X formed by uniformly scattered random grains Ξ ⊂ R ν . We obtain general conditions on the generic grain set Ξ under which p ̂ λ , A has an α -stable limit distribution with index 1 < α ≤ 2 . A large class of Boolean models with randomly homothetic grains satisfying these conditions is introduced. We also discuss the limit distribution of the sample volume fraction of a Boolean set observed on a large subset of a ν 0 -dimensional hyperplane of R ν ( 1 ≤ ν 0 ≤ ν − 1 ). Similar results are also obtained for more general excursion sets of Boolean models.