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Boolean Valued Representation of Random Sets and Markov Kernels with Application to Large Deviations

Author

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  • Antonio Avilés López

    (Departamento de Matematica, Universidad de Murcia, Espinardo, 30100 Murcia, Spain)

  • José Miguel Zapata García

    (School of Mathematics and Statistics, University College Dublin, Belfield, 58622 Dublin 4, Ireland)

Abstract

We establish a connection between random set theory and Boolean valued analysis by showing that random Borel sets, random Borel functions, and Markov kernels are respectively represented by Borel sets, Borel functions, and Borel probability measures in a Boolean valued model. This enables a Boolean valued transfer principle to obtain random set analogues of available theorems. As an application, we establish a Boolean valued transfer principle for large deviations theory, which allows for the systematic interpretation of results in large deviations theory as versions for Markov kernels. By means of this method, we prove versions of Varadhan and Bryc theorems, and a conditional version of Cramér theorem.

Suggested Citation

  • Antonio Avilés López & José Miguel Zapata García, 2020. "Boolean Valued Representation of Random Sets and Markov Kernels with Application to Large Deviations," Mathematics, MDPI, vol. 8(10), pages 1-23, October.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:10:p:1848-:d:431730
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    References listed on IDEAS

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    1. Asgar Jamneshan & Michael Kupper & José Miguel Zapata-García, 2020. "Parameter-Dependent Stochastic Optimal Control in Finite Discrete Time," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 644-666, August.
    2. Ilya Molchanov & Ignacio Cascos, 2016. "Multivariate Risk Measures: A Constructive Approach Based On Selections," Mathematical Finance, Wiley Blackwell, vol. 26(4), pages 867-900, October.
    3. Yuri Kabanov, 2009. "Markets with Transaction Costs. Mathematical Theory," Post-Print hal-00488168, HAL.
    4. Andreas Haier & Ilya Molchanov, 2019. "Multivariate risk measures in the non-convex setting," Papers 1902.00766, arXiv.org, revised Sep 2019.
    5. Molchanov,Ilya & Molinari,Francesca, 2018. "Random Sets in Econometrics," Cambridge Books, Cambridge University Press, number 9781107121201, October.
    6. Ignacio Cascos & Ilya Molchanov, 2013. "Multivariate risk measures: a constructive approach based on selections," Papers 1301.1496, arXiv.org, revised Jul 2016.
    7. Emmanuel Lepinette, 2020. "Random optimization on random sets," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 91(1), pages 159-173, February.
    8. Yuan, Demei & Hu, Xuemei, 2015. "A conditional version of the extended Kolmogorov–Feller weak law of large numbers," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 99-107.
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