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The complete mixability and convex minimization problems with monotone marginal densities. (English) Zbl 1229.60019
It is shown in this paper that distributions \(P\) with a monotone density on a bounded interval are \(n\)-mixable under a moderate mean conditions, i.e., there exist \(X_i \sim P\), \(1 \leq i \leq n\), with \(\sum_{i=1}^{n} X_i =c\). In the case of symmetric unimodal distributions this property was shown before in [L. Rüschendorf and L. Uckelmann, in: Distributions with given marginals and statistical modelling. Papers presented at the meeting, Barcelona, Spain, July 17–20, 2000. Dordrecht: Kluwer Academic Publishers. 211–222 (2002; Zbl 1142.62316)]. Based on this property the generalized variance minimization problem to minimize \(\operatorname{E} f (X_i + \dots + X_n)\) over \(X_i \sim P\) for any convex function \(f\) is solved for distributions \(P\) with monotone density.

MSC:
60E05 Probability distributions: general theory
60E15 Inequalities; stochastic orderings
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