Stochastic Inequalities on Partially Ordered Spaces

In this paper we discuss characterizations, basic properties and applications of a partial ordering, in the set of probabilities on a partially ordered Polish space E, defined by P₁ P₂ iff ∫ f dP₁ ∫ f dP₂ for all real bounded increasing f. A result of Strassen implies that P₁ P₂ is equivalent to the existence of E-valued random variables X₁ X₂ with distributions P₁ and P₂. After treating similar characterizations we relate the convergence properties of P₁ P₂ ⋯ to those of the associated X₁ X₂ ⋯. The principal purpose of the paper is to apply the basic characterization to the problem of comparison of stochastic processes and to the question of the computation of the d̄-distance (defined by Ornstein) of stationary processes. In particular we get a generalization of the comparison theorem of O'Brien to vector-valued processes. The method also allows us to treat processes with continuous time parameter and with paths in D 0, 1.