  Hadar and Russell (1974) and Levy and Paroush (1974) presented sufficient
conditions for multivariate stochastic dominance when the distributions
involved are continuous with compact support. Further generalizations involved
either independence assumptions (Sacarsini (1988)) or the introduction of new
concepts like 'correlation increasing transformation' (Epstein and Tanny
(1980), Tchen (1980), Mayer (2013)). In this paper, we present a direct proof
that extends the original results to the general case where the involved
distributions are only assumed to have compact support. This result has in turn
proven useful for statistical tests of dominance without the assumption of
absolute continuity. The first section introduces several concepts used
throughout the paper. In the second section we recall the classic result as
presented in Atkinson and Bourguignon (1982), with a slightly lighter proof
using the general integration by parts formula for n dimensional
Lebesgue-Stieljes integrals. In the third section we present our proof of the
general result, using Riemman-Stieljes partial sums in a direct fashion that
helps to clarify the role of modularity conditions and boundary effects in the
sufficiency of the conditions. The last section discusses the relevance of the
result and concludes.
