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Large Deviations of Continuous Regular Conditional Probabilities

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  • W. Zuijlen

    (Leiden University)

Abstract

We study product regular conditional probabilities under measures of two coordinates with respect to the second coordinate that are weakly continuous on the support of the marginal of the second coordinate. Assuming that there exists a sequence of probability measures on the product space that satisfies a large deviation principle, we present necessary and sufficient conditions for the conditional probabilities under these measures to satisfy a large deviation principle. The arguments of these conditional probabilities are assumed to converge. A way to view regular conditional probabilities as a special case of product regular conditional probabilities is presented. This is used to derive conditions for large deviations of regular conditional probabilities. In addition, we derive a Sanov-type theorem for large deviations of the empirical distribution of the first coordinate conditioned on fixing the empirical distribution of the second coordinate.

Suggested Citation

  • W. Zuijlen, 2018. "Large Deviations of Continuous Regular Conditional Probabilities," Journal of Theoretical Probability, Springer, vol. 31(2), pages 1058-1096, June.
    • Handle: RePEc:spr:jotpro:v:31:y:2018:i:2:d:10.1007_s10959-016-0733-1
    DOI: 10.1007/s10959-016-0733-1
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    References listed on IDEAS

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    1. den Hollander, F. & Redig, F. & van Zuijlen, W., 2015. "Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 371-400.
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