A subexponential version of Cramer's theorem

TL;DR

This paper extends Cramer's large deviations theorem to subexponential scales, revealing non-convex rate functions and developing new tilting techniques for empirical means of i.i.d. variables.

Contribution

It introduces a subexponential large deviations framework with a novel tilting strategy and characterizes the associated non-convex rate function.

Findings

Non-trivial deviations occur at subexponential scales.

The rate function is non-convex and not from a Legendre-Fenchel transform.

A new tilting method is developed for the lower bound.

Abstract

We consider the large deviations associated with the empirical mean of independent and identically distributed random variables under a subexponential moment condition. We show that non-trivial deviations are observable at a subexponential scale in the number of variables, and we provide the associated rate function, which is non-convex and is not derived from a Legendre-Fenchel transform. The proof adapts the one of Cramer's theorem to the case where the fluctuation is generated by a single variable. In particular, we develop a new tilting strategy for the lower bound, which leads us to introduce a condition on the second derivative of the moment generating function. Our results are illustrated by a couple of simple examples.