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Cramér’s Theorem in Banach Spaces Revisited

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Séminaire de Probabilités XLIX

Part of the book series: Lecture Notes in Mathematics ((SEMPROBAB,volume 2215))

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Abstract

The text summarizes the general results of large deviations for empirical means of independent and identically distributed variables in a separable Banach space, without the hypothesis of exponential tightness. The large deviation upper bound for convex sets is proved in a nonasymptotic form; as a result, the closure of the domain of the entropy coincides with the closed convex hull of the support of the common law of the variables. Also a short original proof of the convex duality between negentropy and pressure is provided: it simply relies on the subadditive lemma and Fatou’s lemma, and does not resort to the law of large numbers or any other limit theorem. Eventually a Varadhan-like version of the convex upper bound is established and embraces both results.

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Notes

  1. 1.

    Hypothesis (\(\hat {\mathbf {C}}\)) of [14] and Assumption 6.1.2 (b) of [13] were introduced so as to complete the proofs of Appendix of [4], but they are not required to prove the first proposition of the appendix.

  2. 2.

    Physically speaking, the function p should be interpreted as the opposite of a free energy, which is proportional to the pressure in the case of simple fluids.

  3. 3.

    The proof is even simpler using the closed half-space upper bound, which is a particular case of (UB cc).

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Acknowledgements

I would like to thank Raphaël Cerf and Yann Fuchs for their careful reading, and the referee for his suggestions.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, UPS IMT, Toulouse Cedex, France

    Pierre Petit

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  1. Pierre Petit
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Correspondence to Pierre Petit .

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Editors and Affiliations

  1. Laboratoire de Mathématiques de Versailles, Université Versailles Saint-Quentin, Versailles, France

    Catherine Donati-Martin

  2. Institut Elie Cartan de Lorraine, Vandoeuvre-les-Nancy, France

    Antoine Lejay

  3. Laboratoire de Mathématiques de Versailles, Université Versailles Saint-Quentin, Versailles, France

    Alain Rouault

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Petit, P. (2018). Cramér’s Theorem in Banach Spaces Revisited. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIX. Lecture Notes in Mathematics(), vol 2215. Springer, Cham. https://doi.org/10.1007/978-3-319-92420-5_12

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