On the existence of probability measures with given marginals

Edwards, David Albert

Annales de l'Institut Fourier, Tome 28 (1978) no. 4, pp. 53-78.

Soit $X$ un espace compact ordonné et soient $\mu ,\nu$ deux probabilités sur $X$ telles que $\mu \left(f\right)\le \nu \left(f\right)$ pour toute fonction croissante continue $f:X\to \mathbf{R}$. Alors nous démontrons qu’il existe une probabilité $\theta$ sur $X\times X$ telle que :

(i) $\theta \left(R\right)=1$, où $R$ est le graphe de l’ordre sur $X$,

(ii) les projections de $\theta$ sur $X$ sont $\mu$ et $\nu$.

On généralise ce théorème aux espaces complètement réguliers ordonnés de Nachbin et, en plus, à certains produits infinis. On met en évidence les relations entre ces résultats et les travaux de Nachbin, Strassen et Hommel.

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 $X\times X$ 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.

[1] A. Badrikian, Séminaire sur les fonctions aléatoires linéaires et les mesures cylindriques, Springer-Verlag, Berlin, 1970. | MR | Zbl

[2] D. A. Edwards, Choquet boundary theory for certain spaces of lower semicontinuous functions, in Function algebras, Scott Foresman and Co., Chicago 1966. | MR | Zbl

[3] D. A. Edwards, Measures on product spaces and the Holley-Preston inequalities, Bull. Lond. Math. Soc., 8 (1976), 7.

[4] D. A. Edwards, On the Holley-Preston inequalities, to appear in Proc. Roy. Soc. of Edinburgh, Section A (Mathematics). | Zbl

[5] G. Hommel, Increasing Radon measures on locally compact ordered spaces, Rendiconti di Matematica, 9 (1976), 85-117. | MR | Zbl

[6] J. H. B. Kemperman, On the FKG-inequality for measures on a partially ordered space (to appear). | Zbl

[7] L. Nachbin, Topology and order, van Nostrand, Princeton, 1965. | Zbl

[8] C. J. Preston, A generalization of the FKG inequalities, Commun. Math. Phys., 36 (1974), 233-241.

[9] H. A. Priestley, Separation theorems for semi-continuous functions on normally ordered topological spaces, J. Lond. Math. Soc., 3 (1971), 371-377. | MR | Zbl

[10] V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist., 36 (1965), 432-439. | MR | Zbl

[11] G. F. Vincent-Smith, Filtering properties of wedges of affine functions, Journ. Lond. Math. Soc., 8 (1974), 621-629. | MR | Zbl