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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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    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.
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