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Cramér’s Theorem in Banach Spaces Revisited

  • Pierre Petit
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2215)

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.

Keywords

Cramér’s theory Large deviations Subadditivity Convexity Fenchel-Legendre transformation 

MSC 2010 Subject Classifications

60F10 

Notes

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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut de Mathématiques de ToulouseUPS IMTToulouse CedexFrance

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