On Strassen’s Theorem on Stochastic Domination

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 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 is a partial ordering; the proof reveals that a pre-ordering suffices.