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

  • Tomasz Schreiber (a1)

Abstract

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.

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Corresponding author

Postal address: Faculty of Mathematics and Computer Science, Nicholas Copernicus University, Ul Chopina 12/18, 87-100 Toruń, Poland. Email address: tomeks@mat.uni.torun.pl

References

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

  • Tomasz Schreiber (a1)

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