Separation Theorems for Increasing Convex and Weak 3-Convex Orders on Real-Valued Random Variables

We examine separation theorems for the increasing convex (stop-loss) order and weak 3-convex order. We improve prior proofs for the increasing convex order by using integrated survival functions. We extend the separation theorem for weak 3-convex order from non-negative random variables to general real-valued random variables with finite second moments via a constructive approach based on higher-order integrated survival functions. A counterexample verifies that the $3$-convex order and $2$-convex order chain does not hold for the weak $3$-convex order. Our results enrich stochastic order theories for actuarial science and risk management.