Stochastic Dominance and Moments of Distributions

Stochastic dominance orders of all finite degrees are defined on the set of distribution functions on the nonnegative real numbers in terms of integrals of the distributions. It is proved that if F strictly nth-degree stochastically dominates G, and if the moments of F and G through order n are finite with μ_F^k = ∫X^kdF (x), then (μ_F¹, …, μ_Fⁿ) ≠ (μ_G¹, …, μ_Gⁿ) and (−1)^k−1 μ_F^k > (−1)^k−1 μ_G^k for the smallest k for which μ_F^k ≠ μ_G^k.