Cram\'er's theorem for asymptotically decoupled fields

TL;DR

This paper extends Cramér's large deviations theorem to a broad setting including i.i.d. vectors and asymptotically decoupled measures, unifying several large deviations results.

Contribution

It establishes a general framework for Cramér's theorem applicable to fields of random vectors, encompassing both i.i.d. and asymptotically decoupled cases.

Findings

Unified large deviations framework for empirical means of random vector fields.

Includes Cramér's theorem for i.i.d. vectors.

Encompasses Sanov's theorem for asymptotically decoupled measures.

Abstract

We give a general setting for Cram\'er's large deviations theorem for the empirical means of a field of random vectors, which contains Cram\'er's theorem for i.i.d. random vectors and Sanov's theorem for asymptotically decoupled measures. ----- Nous \'etablissons un cadre g\'en\'eral pour le th\'eor\`eme de Cram\'er sur les grandes d\'eviations des moyennes empiriques d'un champ de vecteurs al\'eatoires, cadre qui contient le th\'eor\`eme de Cram\'er pour des vecteurs al\'eatoires i.i.d. et le th\'eor\`eme de Sanov pour les mesures asymptotiquement d\'ecoupl\'ees.