An extension of Sanov's theorem: application to the Gibbs conditioning principle

Abstract

A large-deviation principle is proved for the empirical measures of independent and ident\-ically distributed random variables with a topology based on functions having only some exponential moments. The rate function differs from the usual relative entropy: it involves linear forms which are no longer measures. Following Stroock and Zeitouni, the Gibbs conditioning principle (GCP) is then derived with the help of the previous result. Apart from a more direct proof than has previously been available, the main improvements with respect to GCPs already published are the following: convergence holds in situations where the underlying log-Laplace transform (the pressure) may not be steep and the constraints are built on energy functions admitting only some finite exponential moments. Basic techniques from the theory of Orlicz spaces appear to be a powerful tool.