Expected convex hulls, order statistics, and Banach space probabilities. (English) Zbl 0641.60015
The following result is shown: if \(X_ 1,X_ 2,..\). are i.i.d. points in \(R^ d\) with finite expected norm then their common distribution is determined by the sequence \(K_ 1\subseteq K_ 2\subseteq..\). where \(K_ n\) is the expectation of the convex hull of \(X_ 1,...,X_ n.\)
This generalizes a scalar result that goes back to W. Hoeffding [Ann. Math. Stat. 24, 93-100 (1953; Zbl 0050.136)]. Remarks include an application to Gaussian measures on Banach spaces.
This generalizes a scalar result that goes back to W. Hoeffding [Ann. Math. Stat. 24, 93-100 (1953; Zbl 0050.136)]. Remarks include an application to Gaussian measures on Banach spaces.
Reviewer: R.A.Vitale
MSC:
| 60D05 | Geometric probability and stochastic geometry |
| 60E05 | Probability distributions: general theory |
| 52A22 | Random convex sets and integral geometry (aspects of convex geometry) |
| 62E10 | Characterization and structure theory of statistical distributions |
Keywords:
abstract Wiener space; convex hull; order statistic; random sets; Wiener measure; Gaussian measures on Banach spacesCitations:
Zbl 0050.136References:
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