Chapitre D'ouvrage Année : 2018

Cramér’s Theorem in Banach Spaces Revisited

Résumé

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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hal-01976961 , version 1 (10-01-2019)

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P Petit. Cramér’s Theorem in Banach Spaces Revisited. Catherine Donati-Martin, Antoine Lejay, Alain Rouault. Séminaire de Probabilités XLIX., 2215, Springer International Publishing, pp.455-473, 2018, Lecture Notes in Mathematics, 978-3-319-92419-9. ⟨10.1007/978-3-319-92420-5_12⟩. ⟨hal-01976961⟩
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