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Sanov's theorem in the Wasserstein distance: A necessary and sufficient condition

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  • Wang, Ran
  • Wang, Xinyu
  • Wu, Liming

Abstract

Let (Xn)n>=1 be a sequence of i.i.d.r.v.'s with values in a Polish space of law [mu]. Consider the empirical measures . Our purpose is to generalize Sanov's theorem about the large deviation principle of Ln from the weak convergence topology to the stronger Wasserstein metric Wp. We show that Ln satisfies the large deviation principle in the Wasserstein metric Wp (p[set membership, variant][1,+[infinity])) if and only if for all [lambda]>0, and for some x0[set membership, variant]E.

Suggested Citation

  • Wang, Ran & Wang, Xinyu & Wu, Liming, 2010. "Sanov's theorem in the Wasserstein distance: A necessary and sufficient condition," Statistics & Probability Letters, Elsevier, vol. 80(5-6), pages 505-512, March.
    • Handle: RePEc:eee:stapro:v:80:y:2010:i:5-6:p:505-512
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    References listed on IDEAS

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    1. Eichelsbacher, Peter & Schmock, Uwe, 1998. "Exponential approximations in completely regular topological spaces and extensions of Sanov's theorem," Stochastic Processes and their Applications, Elsevier, vol. 77(2), pages 233-251, September.
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    Cited by:

    1. Deuschel, Jean-Dominique & Friz, Peter K. & Maurelli, Mario & Slowik, Martin, 2018. "The enhanced Sanov theorem and propagation of chaos," Stochastic Processes and their Applications, Elsevier, vol. 128(7), pages 2228-2269.
    2. Ran Ji & Miguel A. Lejeune, 2021. "Data-Driven Optimization of Reward-Risk Ratio Measures," INFORMS Journal on Computing, INFORMS, vol. 33(3), pages 1120-1137, July.
    3. Liu, Wei & Wu, Liming, 2020. "Large deviations for empirical measures of mean-field Gibbs measures," Stochastic Processes and their Applications, Elsevier, vol. 130(2), pages 503-520.
    4. Gao, Fuqing & Wang, Shaochen, 2011. "Asymptotic behavior of the empirical conditional value-at-risk," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 345-352.

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