On distributions of random closed sets and expected convex hulls
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 \(C(S^*))\).
scientific article; zbMATH DE number 445305