Implicit Markov kernels in probability theory
Hlubinka, Daniel
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002), p. 547-564 / Harvested from Czech Digital Mathematics Library
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Having Polish spaces $\Bbb X$, $\Bbb Y$ and $\Bbb Z$ we shall discuss the existence of an $\Bbb X \times \Bbb Y$-valued random vector $(\xi,\eta )$ such that its conditional distributions $\operatorname{K}_{x} = \Cal L(\eta\mid \xi=x)$ satisfy $e(x, \operatorname{K}_{x}) = c(x)$ or $e(x,\operatorname{K}_{x}) \in C(x)$ for some maps $e:\Bbb X\times \Cal M_1(\Bbb Y) \to \Bbb Z$, $c:\Bbb X \to \Bbb Z$ or multifunction $C:\Bbb X \to 2^{\Bbb Z}$ respectively. The problem is equivalent to the existence of universally measurable Markov kernel $\operatorname{K}:\Bbb X \to \Cal M_1(\Bbb Y)$ defined implicitly by $e(x, \operatorname{K}_{x}) = c(x)$ or $e(x,\operatorname{K}_{x}) \in C(x)$ respectively. In the paper we shall provide sufficient conditions for the existence of the desired Markov kernel. We shall discuss some special solutions of the $(e,c)$- or $(e,C)$-problem and illustrate the theory on the generalized moment problem.

Publié le : 2002-01-01
Classification:  28A35,  28B20,  46A55,  60A10,  60B05 
@article{119345,
     author = {Daniel Hlubinka},
     title = {Implicit Markov kernels in probability theory},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     volume = {43},
     year = {2002},
     pages = {547-564},
     zbl = {1091.28003},
     mrnumber = {1920531},
     language = {en},
     url = {http://dml.mathdoc.fr/item/119345}
}

Hlubinka, Daniel. Implicit Markov kernels in probability theory. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 547-564. http://gdmltest.u-ga.fr/item/119345/

Aubin J.-A.; Frankowska H. Set-valued Analysis, Birkhäuser, Boston, 1990. | MR 1048347 | Zbl 1168.49014

Cohn D.L. Measure Theory, Birkhäuser, Boston, second edition, 1993. | MR 1454121 | Zbl 0860.28001

Kallenberg O. Foundations of modern probability, Probab. Appl., Springer Verlag, New York, 1997. | MR 1464694 | Zbl 0996.60001

Srivastava S.M. A Course on Borel Sets, Graduate Texts in Mathematics, vol. 180, Springer Verlag, New York, 1998. | MR 1619545 | Zbl 0903.28001

Štěpán J.; Hlubinka D. Two-dimensional probabilities with a given conditional structure, Kybernetika 35(3) 367-381 (1999). (1999) | MR 1704672

Štěpán J. How to construct a two-dimensional random vector with a given conditional structure, in: Viktor Beneš and Josef Štěpán, Eds, {Distribution with given marginals and moment problems}, 1997. | MR 1614669

Topsøe F. Topology and Measure, Lecture Notes in Mathematics 133, Springer Verlag, Berlin, 1970. | MR 0422560

Winkler G. Choquet Order and Simplices, Lecture Notes in Mathematics 1145, Springer Verlag, Berlin, 1985. | MR 0808401 | Zbl 0578.46010

Winkler G. Extreme points of moment sets, Mathematics of operational research 13(4) 581-587 (1988). (1988) | MR 0971911 | Zbl 0669.60009