Cram\'er's theorem in measurable locally convex spaces

TL;DR

This paper develops a unified framework for Cramér's large deviations theorem applicable to empirical means of i.i.d. random vectors in measurable locally convex spaces, extending classical results in Banach spaces.

Contribution

It introduces a general setting that encompasses Cramér's theorem in Banach spaces and Sanov's theorem within locally convex spaces.

Findings

Unified large deviations framework for locally convex spaces

Extension of Cramér's theorem beyond Banach spaces

Includes Sanov's theorem as a special case

Abstract

We give a general setting for Cram\'er's large deviations theorem for the empirical means of a sequence of i.i.d. random vectors, which contains Cram\'er's theorem in a Banach space and Sanov's theorem. ----- Nous \'etablissons un cadre g\'en\'eral pour le th\'eor\`eme de Cram\'er sur les grandes d\'eviations des moyennes empiriques d'une suite de vecteurs al\'eatoires i.i.d., cadre qui contient le th\'eor\`eme de Cram\'er dans les espaces de Banach s\'eparables et le th\'eor\`eme de Sanov.