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Keywords:
two-dimensional probabilities; extremal measure
two-dimensional probabilities; extremal measure
Summary:
A properly measurable set ${\cal P} \subset {X} \times M_1({Y})$ (where ${X}, {Y}$ are Polish spaces and $M_1(Y)$ is the space of Borel probability measures on $Y$) is considered. Given a probability distribution $\lambda \in M_1(X)$ the paper treats the problem of the existence of ${X}\times {Y}$-valued random vector $(\xi ,\eta )$ for which ${\cal L}(\xi )=\lambda $ and ${\cal L}(\eta | \xi =x) \in {\cal P}_x$ $\lambda $-almost surely that possesses moreover some other properties such as “${\cal L}(\xi ,\eta )$ has the maximal possible support” or “${\cal L}(\eta | \xi =x)$’s are extremal measures in ${\cal P}_x$’s”. The paper continues the research started in [7].
References:
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[2] Beneš V., (eds.) J. Štěpán: Distributions with Given Marginals and Moment Problems. Kluwer, Dordrecht 1997 MR 1614650 | Zbl 0885.00054
[3] Cohn D. L.: Measure Theory. Birkhäuser, Boston 1980 MR 0578344 | Zbl 0860.28001
[4] Kempermann J. H. B.: The general moment problem, a geometric approach. Ann. Math. Statist. 39 (1968), 93–122 DOI 10.1214/aoms/1177698508 | MR 0247645
[5] Meyer P. A.: Probability and Potentials. Blaisdell, Waltham 1966 MR 0205288 | Zbl 0271.60086
[6] Schwarz L.: Radon Measures on Arbitrary Topological Spaces and Cylindrical Measures. Oxford University Press, Oxford 1973 MR 0426084
[8] Winkler G.: Choquet Order and Simplices. (Lectures Notes in Mathematics 1145.) Springer–Verlag, Berlin 1985 MR 0808401 | Zbl 0578.46010
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