The complete mixability and convex minimization problems with monotone marginal densities

Following the results of Rüschendorf and Uckelmann (2002) [20], we introduce the completely mixable distributions on and prove that the distributions with monotone density and moderate mean are completely mixable. Using this method, we solve the minimization problem 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.