## The Annals of Probability

- Ann. Probab.
- Volume 5, Number 6 (1977), 899-912.

### Stochastic Inequalities on Partially Ordered Spaces

T. Kamae, U. Krengel, and G. L. O'Brien

#### 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 $D\lbrack 0, 1\rbrack$.

#### Article information

**Source**

Ann. Probab., Volume 5, Number 6 (1977), 899-912.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995659

**Digital Object Identifier**

doi:10.1214/aop/1176995659

**Mathematical Reviews number (MathSciNet)**

MR494447

**Zentralblatt MATH identifier**

0371.60013

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60B99: None of the above, but in this section

Secondary: 60G99: None of the above, but in this section 60G10: Stationary processes

**Keywords**

Stochastic comparison measures on partially ordered spaces stationary processes $\bar d$-distance

#### Citation

Kamae, T.; Krengel, U.; O'Brien, G. L. Stochastic Inequalities on Partially Ordered Spaces. Ann. Probab. 5 (1977), no. 6, 899--912. doi:10.1214/aop/1176995659. https://projecteuclid.org/euclid.aop/1176995659