Cramér's theorem

Cramér's theorem: Let $(X_i)_{i\ge 1}$ be an i.i.d. sequence of real-valued random variables. Assume the moment generating function $M(\theta)=\mathbb{E}[e^{\theta X_1}]$ is finite for all $\theta$ in some open interval containing $0$, and let $\Lambda(\theta)=\log M(\theta)$ be the log moment generating function. Define the empirical mean $\overline X_n=\frac{1}{n}\sum_{i=1}^n X_i$. Then $(\overline X_n)$ satisfies a large deviation principle on $\mathbb{R}$ with speed $n$ and good rate function

The rate function $I$ is the Cramér transform, i.e. the Legendre–Fenchel transform (Fenchel conjugate) of $\Lambda$.