The complete mixability and convex minimization problems with monotone marginal densities

Following the results in Rüschendorf and Uckelmann (2002), we introduce the completely mixable distributions on R and prove that distributions with monotone density and moderate mean are completely mixable. Using this method we solve the minimization problem minXi∼P Ef(X1 + · · ·+ Xn) for convex functions f and marginal distributions P with monotone density. Our results also provide valuable implications in variance minimization, bounds for the sum of random variables and risk theory. Key-words: complete mixability; variance minimization; multivariate dependence; monotone densities; optimal coupling.