  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.
