The complete mixability and convex minimization problems with monotone marginal densities (Q634547)

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 [\textit{L. Rüschendorf} and \textit{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.