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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Monotone bivariate Markov kernels with specified marginals


Authors: Motoya Machida and Alexander Shibakov
Journal: Proc. Amer. Math. Soc. 138 (2010), 2187-2194
MSC (2010): Primary 60E15; Secondary 62E15, 60J05
Published electronically: February 1, 2010
MathSciNet review: 2596058
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Abstract: Given two Markov kernels $ k$ and $ k'$ on an ordered Polish space, such that $ k$ is stochastically dominated by $ k'$, we establish the existence of: (i) a monotone bivariate Markov kernel whose marginals are $ k$ and $ k'$ and (ii) an upward coupler from $ k$ to $ k'$. This extends the results of Strassen, Kamae, Krengel and O'Brien to Markov kernels. Two examples are also given. The first is a simple illustration of our original motivation for this work, while the second demonstrates the optimality of our main result. The key technique is a combination of the standard probability/charge approach and the use of measurable selections of multivalued measurable maps.


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Additional Information

Motoya Machida
Affiliation: Department of Mathematics, Tennessee Technological University, Cookeville, Tennessee 38505
Email: mmachida@tntech.edu

Alexander Shibakov
Affiliation: Department of Mathematics, Tennessee Technological University, Cookeville, Tennessee 38505
Email: alex@math.tntech.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10241-X
PII: S 0002-9939(10)10241-X
Keywords: Probability measures with given marginals, stochastically monotone Markov kernel, measurable selection, coupling, Fill's algorithm
Received by editor(s): February 9, 2009
Received by editor(s) in revised form: October 5, 2009
Published electronically: February 1, 2010
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.