On distributions of random closed sets and expected convex hulls
type: Item
instance of: scholarly article
title: On distributions of random closed sets and expected convex hulls
zbMATH Open document ID: 0789.60009
DOI: 10.1016/0167-7152(93)90199-S
author: Ilya S. Molchanov
published in: Statistics \& Probability Letters
publication date: 1994-06-06T00:00:00Z
review text: As a generalization of a classical result on order statistics, \textit{R. A. Vitale} [Acta Appl. Math. 9, 97-102 (1987; Zbl 0641.60015)] showed that the common distribution of an i.i.d. sequence \(X_ 1,X_ 2,\dots\) of random vectors (with existing expectation) in a separable Banach space \(\mathbb{B}\) is determined by the nested sequence of (convex, compact) moment bodies, that is, by the set-valued expectations \((\mathbb{E} \text{conv} \{X_ 1,\dots,X_ n\})^ \infty_{n=1}\) of the convex hulls \(\text{conv} \{X_ 1,\dots,X_ n\}\). This result is extended to random convex compact sets \(A_ 1,A_ 2,\dots\) in \(\mathbb{B}\) by applying it to the Banach space \(C(S^*)\) of continuous functions on the unit sphere \(S^*\) in the dual \(\mathbb{B}^*\) of \(\mathbb{B}\). The key is the embedding of complex compact subsets of \(\mathbb{B}\) into \(C(S^*)\) via the support function. The ``moment bodies'' are, in this extension, expected convex hulls of support functions (as elements of \(C(S^*))\).
Mathematics Subject Classification ID: 60D05
Mathematics Subject Classification ID: 60B11
zbMATH DE Number: 445305
zbMATH Keywords: random closed set
zbMATH Keywords: convex hull
zbMATH Keywords: support function
zbMATH Keywords: expectation
zbMATH Keywords: distribution
reviewed by: Wolfgang Weil
MaRDI profile type: Publication
cites work: A strong law of large numbers for random compact sets
cites work: Integrals of set-valued functions
cites work: Characterizations of probability distributions. A unified approach with an emphasis on exponential and related models
cites work: Q4084463
cites work: On some partial orderings of random closed sets
cites work: Expected convex hulls, order statistics, and Banach space probabilities
cites work: An application of the central limit theorem for banach-space-valued random variables to the theory of random sets
full work available at URL: 0167-7152(93)90199-s
OpenAlex ID: W2019367290
description: scientific article; zbMATH DE number 445305