Given $X_1,...$ of iid random variables. We know that if the moment generating function $M(\theta) < \infty, \forall \theta $ from Cramérs Theorem we get:
$\lim_{n\to \infty} \frac{1}{n}\log \mathbb{P}(S_n \ge na) = -I(a)$ where
$I(a) = \sup_\theta(a\theta - \log M(\theta))$.
Question: What happens if $M(\theta)$ isn't finite for all values of $\theta$? Namely, is there another version of Cramérs theorem to help when calculating the rate function $I$ when this situation comes up.
Yes. Let $\Theta$ be the set of $\theta$ such that $M(\theta)<\infty$. It is easy to check that $\Theta$ is convex, that is, an interval. As long as $\theta$ is in the interior of $\Theta$, and solves $M'(\theta)/M(\theta)=a$, that is, the maximization problem defining $I(a)$ is solved the calculus way, the conclusion holds for that value of $a$. The condition that you cite, that $\Theta=\mathbb R$, is introduced for convenience of exposition.
I'm away from my books at the moment, so I can't give a reference to this version.
