Variance asymptotics and central limit theorems for volumes of unions of random closed sets
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 ∪ ∙ ∙ ∙ ∪ Xn)) 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 Rd with centres distributed according to a spherically-symmetric heavy-tailed law.
Permanent link to this document: http://projecteuclid.org/euclid.aap/1033662164
Digital Object Identifier: doi:10.1239/aap/1033662164
Mathematical Reviews number (MathSciNet): MR1929596
Zentralblatt MATH identifier: 1018.60012