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 R and prove that the distributions with monotone density and moderate mean are completely mixable. Using this method, we solve the minimization problem min X i ∼ P E f ( X 1 + ⋯ + X n ) 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.