Variance asymptotics and central limit theorems for volumes of unions of random closed sets

Abstract

Let X, X₁ ,X₂ ,... be a sequence of i.i.d. random closed subsets of a certain locally compact, Hausdorff and separable base space E. For a fixed normalised Borel measure μ on E, we investigate the behaviour of random variables μ(E \ (X₁ ∪ ∙ ∙ ∙ ∪ X_n)) for large n. The results obtained include a description of variance asymptotics, strong law of large numbers and a central limit theorem. As an example we give an application of the developed methods for asymptotic analysis of the mean width of convex hulls generated by uniform samples from a multidimensional ball. Another example deals with unions of random balls in R^d with centres distributed according to a spherically-symmetric heavy-tailed law.