This article is cited in 11 papers

Limit theorems for certain functionals of unions of random closed sets

T. Schreiber

Nikolaus Copernicus University, Faculty of Mathematics and Informatics

Abstract: Let $X_1,X_2,\dots$ 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 $Q_Y(X_1\cup\dots\cup X_n)$. 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.

Received: 06.10.1999

Language: English

DOI: 10.4213/tvp3004

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

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