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 less than or similar to 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 less than or similar to is a partial ordering; the proof reveals that a pre-ordering suffices.