Open Access
December, 1977 Stochastic Inequalities on Partially Ordered Spaces
T. Kamae, U. Krengel, G. L. O'Brien
Ann. Probab. 5(6): 899-912 (December, 1977). DOI: 10.1214/aop/1176995659


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 $D\lbrack 0, 1\rbrack$.


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T. Kamae. U. Krengel. G. L. O'Brien. "Stochastic Inequalities on Partially Ordered Spaces." Ann. Probab. 5 (6) 899 - 912, December, 1977.


Published: December, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0371.60013
MathSciNet: MR494447
Digital Object Identifier: 10.1214/aop/1176995659

Primary: 60B99
Secondary: 60G10 , 60G99

Keywords: $\bar d$-distance , measures on partially ordered spaces , Stationary processes , Stochastic comparison

Rights: Copyright © 1977 Institute of Mathematical Statistics

Vol.5 • No. 6 • December, 1977
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