Multivariate Stochastic Dominance and Moments

The sequence ≥_n^d of nth degree stochastic dominances for d-dimensional distribution functions is defined. It is shown that, under some regularity conditions, ≥_n^d implies ≥_n^d−1 for the (d − 1)-dimensional marginals. Also some necessary conditions for ≥_n^d are established. These conditions, which involve moments, are analogous to the ones proved by Fishburn (1980b) and O'Brien (1984) for univariate distributions.