Stochastic Inequalities on Partially Ordered Spaces

Abstract

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 $P_1 \prec P_2 \operatorname{iff} \int f dP_1\leqq \int f dP_2$ for all real bounded increasing $f$. A result of Strassen implies that $P_1 \prec P_2$ is equivalent to the existence of $E$-valued random variables $X_1 \leqq X_2$ with distributions $P_1$ and $P_2$. After treating similar characterizations we relate the convergence properties of $P_1 \prec P_2 \prec \cdots$ to those of the associated $X_1 \leqq X_2 \leqq \cdots$. 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 $\bar{d}-$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 $D\lbrack 0, 1\rbrack$.