Open Access
December, 1978 Stochastic Partial Ordering
T. Kamae, U. Krengel
Ann. Probab. 6(6): 1044-1049 (December, 1978). DOI: 10.1214/aop/1176995392


A probability measure $P$ on a partially ordered Polish space $E$ is called stochastically smaller than $Q$ (notation: $P \leqslant Q$) if $\int f dP \leqslant \int f dQ$ holds for all bounded increasing measurable $f$. We investigate the question when for a stochastically increasing family $\{P_t, t \in \mathbb{R}\}$ there exists an increasing process $\{X_t, t \in \mathbb{R}\}$ with 1-dimensional marginal distributions $P_t$. A sufficient condition, satisfied, e.g., for $E = \mathbb{R}^\mathbf{N}$, for compact $E$ and for spaces $E$ of Lipschitz-functions, is the compactness of all intervals $\{z \in E: x \leqslant z \leqslant y\}$; but for general countable $E$ such an increasing $E$-valued process $\{X_t\}$ need not exist.


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T. Kamae. U. Krengel. "Stochastic Partial Ordering." Ann. Probab. 6 (6) 1044 - 1049, December, 1978.


Published: December, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0392.60012
MathSciNet: MR512419
Digital Object Identifier: 10.1214/aop/1176995392

Primary: 60B99
Secondary: 60G99

Keywords: increasing processes , Stochastic partial ordering

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 6 • December, 1978
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