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Let  X ,  X  1 ,  X  2 , … 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  \ ( X  1  ∪ ∙ ∙ ∙ ∪  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.

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