# On the existence of probability measures with given marginals

• Volume: 28, Issue: 4, page 53-78
• ISSN: 0373-0956

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## Abstract

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Let $X$ be a compact ordered space and let $\mu ,\nu$ be two probabilities on $X$ such that $\mu \left(f\right)\le \nu \left(f\right)$ for every increasing continuous function $f:X\to \mathbf{R}$. Then we show that there exists a probability $\theta$ on $X×X$ such that(i) $\theta \left(R\right)=1$, where $R$ is the graph of the order in $X$,(ii) the projections of $\theta$ onto $X$ are $\mu$ and $\nu$.This theorem is generalized to the completely regular ordered spaces of Nachbin and also to certain infinite products. We show how these theorems are related to certain results of Nachbin, Strassen and Hommel.

## How to cite

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Edwards, David Alan. "On the existence of probability measures with given marginals." Annales de l'institut Fourier 28.4 (1978): 53-78. <http://eudml.org/doc/74386>.

@article{Edwards1978,
abstract = {Let $X$ be a compact ordered space and let $\mu ,\nu$ be two probabilities on $X$ such that $\mu (f) \le \nu (f)$ for every increasing continuous function $f:X\rightarrow \{\bf R\}$. Then we show that there exists a probability $\theta$ on $X\times X$ such that(i) $\theta (R) =1$, where $R$ is the graph of the order in $X$,(ii) the projections of $\theta$ onto $X$ are $\mu$ and $\nu$.This theorem is generalized to the completely regular ordered spaces of Nachbin and also to certain infinite products. We show how these theorems are related to certain results of Nachbin, Strassen and Hommel.},
author = {Edwards, David Alan},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {4},
pages = {53-78},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the existence of probability measures with given marginals},
url = {http://eudml.org/doc/74386},
volume = {28},
year = {1978},
}

TY - JOUR
AU - Edwards, David Alan
TI - On the existence of probability measures with given marginals
JO - Annales de l'institut Fourier
PY - 1978
PB - Association des Annales de l'Institut Fourier
VL - 28
IS - 4
SP - 53
EP - 78
AB - Let $X$ be a compact ordered space and let $\mu ,\nu$ be two probabilities on $X$ such that $\mu (f) \le \nu (f)$ for every increasing continuous function $f:X\rightarrow {\bf R}$. Then we show that there exists a probability $\theta$ on $X\times X$ such that(i) $\theta (R) =1$, where $R$ is the graph of the order in $X$,(ii) the projections of $\theta$ onto $X$ are $\mu$ and $\nu$.This theorem is generalized to the completely regular ordered spaces of Nachbin and also to certain infinite products. We show how these theorems are related to certain results of Nachbin, Strassen and Hommel.
LA - eng
UR - http://eudml.org/doc/74386
ER -

## References

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1. [1] A. BADRIKIAN, Séminaire sur les fonctions aléatoires linéaires et les mesures cylindriques, Springer-Verlag, Berlin, 1970. Zbl0209.48402MR43 #4994
2. [2] D. A. EDWARDS, Choquet boundary theory for certain spaces of lower semicontinuous functions, in Function algebras, Scott Foresman and Co., Chicago 1966. Zbl0145.38601MR33 #4708
3. [3] D. A. EDWARDS, Measures on product spaces and the Holley-Preston inequalities, Bull. Lond. Math. Soc., 8 (1976), 7.
4. [4] D. A. EDWARDS, On the Holley-Preston inequalities, to appear in Proc. Roy. Soc. of Edinburgh, Section A (Mathematics). Zbl0387.28019
5. [5] G. HOMMEL, Increasing Radon measures on locally compact ordered spaces, Rendiconti di Matematica, 9 (1976), 85-117. Zbl0393.28014MR53 #13504
6. [6] J. H. B. KEMPERMAN, On the FKG-inequality for measures on a partially ordered space (to appear). Zbl0384.28012
7. [7] L. NACHBIN, Topology and order, van Nostrand, Princeton, 1965. Zbl0131.37903
8. [8] C. J. PRESTON, A generalization of the FKG inequalities, Commun. Math. Phys., 36 (1974), 233-241.
9. [9] H. A. PRIESTLEY, Separation theorems for semi-continuous functions on normally ordered topological spaces, J. Lond. Math. Soc., 3 (1971), 371-377. Zbl0207.21203MR43 #3999
10. [10] V. STRASSEN, The existence of probability measures with given marginals, Ann. Math. Statist., 36 (1965), 432-439. Zbl0135.18701MR31 #1693
11. [11] G. F. VINCENT-SMITH, Filtering properties of wedges of affine functions, Journ. Lond. Math. Soc., 8 (1974), 621-629. Zbl0312.46018MR50 #14148

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