A linear version of Dawson-G{\"a}rtner theorem -- Applications to Cram{\'e}r's theory

TL;DR

This paper establishes a linear version of Dawson-Gärtner theorem, showing the preservation of large deviation principles and key equalities through linear projective limits, with applications to Cramér's theory in various spaces.

Contribution

It introduces a linear version of the Dawson-Gärtner theorem, extending the equality between negentropy and Fenchel-Legendre transform to broader spaces and limits.

Findings

Equality --s = p* holds in general for empirical means of i.i.d. variables in normed spaces.

The theorem applies to projective limits of such spaces, broadening its scope.

An example is provided where --s differs from p*, illustrating the limits of the equality.

Abstract

We prove a linear version of Dawson-G{\"a}rtner theorem: weak large deviation principles and the equality --s = p* between the negentropy and the Fenchel-Legendre transform of the pressure are preserved through linear projective limits. As a result, the equality --s = p* holds in great generality for empirical means of independent and identically distributed random variables (Cram{\'e}r's theory), e.g. in any measurable normed space, and even in any projective limit of such spaces. Eventually, we give an original example where --s is different from p* and discuss the dual equality.