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 Teor. Veroyatnost. i Primenen., 2002, Volume 47, Issue 1, Pages 130–142 (Mi tvp3004)

Limit theorems for certain functionals of unions of random closed sets

T. Schreiber

Nikolaus Copernicus University, Faculty of Mathematics and Informatics

Abstract: Let $X_1,X_2,…$ be a sequence of independent identically distributed random closed subsets of a certain locally compact, Hausdorff, and separable space $E$. For each random closed set $Y$ we consider its avoidance functional $Q_Y(F)$ equal to the probability that $Y$ is disjoint with the closed subset $F\subseteq E$. The purpose of this paper is to establish limit theorems for the random variables $Q_Y(X_1\cup…\cup X_n)$. The results obtained are then applied for asymptotic analysis of the mean width of convex hulls generated by uniform samples on a multidimensional ball.

Keywords: random sets, unions of closed sets, hitting functionals, extreme values, convex hulls, mean width, perimeter.

DOI: https://doi.org/10.4213/tvp3004

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English version:
Theory of Probability and its Applications, 2003, 47:1, 79–90

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Citation: T. Schreiber, “Limit theorems for certain functionals of unions of random closed sets”, Teor. Veroyatnost. i Primenen., 47:1 (2002), 130–142; Theory Probab. Appl., 47:1 (2003), 79–90

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp3004
• https://doi.org/10.4213/tvp3004
• http://mi.mathnet.ru/eng/tvp/v47/i1/p130

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Schreiber T., “Variance asymptotics and central limit theorems for volumes of unions of random closed sets”, Adv. in Appl. Probab., 34:3 (2002), 520–539
2. Reitzner M., “Random polytopes and the Efron–Stein jackknife inequality”, Ann. Probab., 31:4 (2003), 2136–2166
3. Schreiber T., “Asymptotic geometry of high-density smooth-grained Boolean models in bounded domains”, Adv. in Appl. Probab., 35:4 (2003), 913–936
4. Calka P., Schreiber T., “Limit theorems for the typical Poisson–Voronoi cell and the Crofton cell with a large inradius”, Ann. Probab., 33:4 (2005), 1625–1642
5. Molchanov I., Theory of Random Sets, 2Nd Edition, Probability Theory and Stochastic Modelling, 87, Springer International Publishing Ag, 2017