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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:
 6e+06 Probability distributions: general theory 6e+16 Inequalities; stochastic orderings
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##### References:
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