Stationary-excess operator and convex stochastic orders

The present paper aims to point out how the stationary-excess operator and its iterates transform s -convex stochastic orders and the associated moment spaces. This allows us to propose a new unified method on constructing s -convex extrema for distributions that are known to be t -monotone. Both discrete and continuous cases are investigated. Several extremal distributions under monotonicity conditions are derived. They are illustrated with some applications in insurance.