Duality and perfect probability spaces

Authors:
D. Ramachandran and L. Rüschendorf

Journal:
Proc. Amer. Math. Soc. **124** (1996), 2223-2228

MSC (1991):
Primary 60A10, 28A35

DOI:
https://doi.org/10.1090/S0002-9939-96-03462-4

MathSciNet review:
1342043

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given probability spaces let denote the set of all probabilities on the product space with marginals and and let be a measurable function on Continuous versions of linear programming stemming from the works of Monge (1781) and Kantorovich-Rubin\v{s}tein (1958) for the case of compact metric spaces are concerned with the validity of the duality

(where is the collection of all probability measures on with and as the marginals). A recently established general duality theorem asserts the validity of the above duality whenever at least one of the marginals is a perfect probability space. We pursue the converse direction to examine the interplay between the notions of duality and perfectness and obtain a new characterization of perfect probability spaces.

**1.**R. M. Dudley,*Probabilities and metrics*, Matematisk Institut, Aarhus Universitet, Aarhus, 1976. Convergence of laws on metric spaces, with a view to statistical testing; Lecture Notes Series, No. 45. MR**0488202****2.**Gnedenko, B.V., and Kolomogorov, A.N. (1954). Limit distributions for sums of independent random variables. Addison Wesley, Cambridge. MR**16:52d****3.**L. V. Kantorovič and G. Š. Rubinšteĭn,*On a space of completely additive functions*, Vestnik Leningrad. Univ.**13**(1958), no. 7, 52–59 (Russian, with English summary). MR**0102006****4.**Hans G. Kellerer,*Duality theorems for marginal problems*, Z. Wahrsch. Verw. Gebiete**67**(1984), no. 4, 399–432. MR**761565**, https://doi.org/10.1007/BF00532047**5.**Monge, G. (1781). Mémoire sur la théorie des déblais ét ramblais.*Mem. Math. Phys. Acad. Roy. Sci. Paris*, 666-704.**6.**Jacques Neveu,*Mathematical foundations of the calculus of probability*, Translated by Amiel Feinstein, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1965. MR**0198505****7.**Jan K. Pachl,*Two classes of measures*, Colloq. Math.**42**(1979), 331–340. MR**567571****8.**Jan K. Pachl,*Correction to: “Two classes of measures” [Colloq. Math. 42 (1979), 331–340; MR 82b:28012]*, Colloq. Math.**45**(1981), no. 2, 331–333. MR**665799****9.**Svetlozar T. Rachev,*Probability metrics and the stability of stochastic models*, Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 1991. MR**1105086****10.**Ramachandran, D. (1979). Perfect Measures, I and II. ISI Lecture Notes Series,**5**and**7**, New Delhi, Macmillan. MR**81h:6005b**; MR**81h:6005a****11.**Ramachandran, D. and Rüschendorf, L. (1995). A general duality theorem for marginal problems.*Probab. Theory Relat. Fields*,**101**, 311-319.**12.**V. Strassen,*The existence of probability measures with given marginals*, Ann. Math. Statist.**36**(1965), 423–439. MR**0177430**, https://doi.org/10.1214/aoms/1177700153

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
60A10,
28A35

Retrieve articles in all journals with MSC (1991): 60A10, 28A35

Additional Information

**D. Ramachandran**

Affiliation:
Department Of Mathematics and Statistics, California State University, 6000 J Street, Sacramento, California 95819-6051

Email:
chandra@csus.edu

**L. Rüschendorf**

Affiliation:
California State University, Sacramento and Universität Freiburg

Address at time of publication:
Institut für Mathematische Stochastik, Albert-Ludwigs-Universität, Hebelstr. 27, D-79104 Freiburg, Germany

Email:
ruschen@buffon.mathematik.uni-freiburg.de

DOI:
https://doi.org/10.1090/S0002-9939-96-03462-4

Keywords:
Duality theorem,
marginals,
perfect measure,
Marczewski function

Received by editor(s):
December 15, 1994

Additional Notes:
Research supported in part by an Internal Awards Grant from the California State University, Sacramento

Communicated by:
Richard T. Durrett

Article copyright:
© Copyright 1996
American Mathematical Society