On Strassen's Theorem on Stochastic Domination

Abstract

The purpose of this note is to make available a reasonably complete and straightforward proof of Strassen's theorem on stochastic domination, and to draw attention to the original paper. We also point out that the maximal possible value of $P(Z = Z')$ is actually not reduced by the requirement $Z \leq Z'$. Here, $Z,Z'$ are stochastic elements that Strassen's theorem states exist under a stochastic domination condition. The consequence of that observation to stochastically monotone Markov chains is pointed out. Usually the theorem is formulated with the assumption that $\leq$ is a partial ordering; the proof reveals that a pre-ordering suffices.