A subexponential version of Cramer's theorem
Probability
2025-07-22 v2 Statistical Mechanics
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.
Cite
@article{arxiv.2206.05791,
title = {A subexponential version of Cramer's theorem},
author = {Grégoire Ferré},
journal= {arXiv preprint arXiv:2206.05791},
year = {2025}
}