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 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. Introduction, preliminariesStrassen's theorem on stochastic domination is now established as a canonical result in probability: it is a crucial tool in the theory of interacting particle systems, cf. Liggett (1985), and has also found many interesting applications in other areas, cf. Preston (1976), Kamae, Krengel andO'Brien (1977), andLindvall (1992). But there seems to be no proof of it easily available. The one in Liggett (1985) and Lindvall (1992) treats only the compact space case, and in Strassen (1965) the result is rather implicit, only briefly mentioned as a special case of a general theorem. This makes a straightforward proof worth our while, considering the importance of the theorem. Strassen (1965) works in a Polish space, i.e., it is (homeomorphic to) a complete and separable metric space. This condition has later been relaxed in several steps, cf. Skala (1993), but the main arguments of the original proof have not been altered. This gives a reason to draw at-51

Mechanism design via optimal transport

“…We now apply a theorem of Strassen for the partial order , using the formulation from [12] and noting that the set {(a, b) : a b} is closed:…”

“…In the proofs of our structural theorems, we employ Strassen's theorem on stochastic domination of measures [12]. As a consequence of our proof technique, we introduce a condition on stochastic domination in both our results.…”

The length of the longest common subsequence of two independent mallows permutations

“…By Strassen's theorem (cf. Lindvall et al (1999)), the conjecture is equivalent to the following statement: there exists a coupling (X, Y ) with X ∼ µ n,q and Y ∼ µ n,q ′ such that X ≤ L Y . 2.…”

Optimal Auctions with Restricted Allocations

“…A matching which satisfies this cell condition will be said to "respect ℓ S ". Once we verify that µ(cell(s)) = 0 for u * , we then show that such a γ * respecting ℓ S exists, either by directly constructing it or by applying Strassen's [1965] Theorem (or more precisely a special case of it [Kamae et al 1977;Lindvall 1999]).…”