= Solution
Let $Y_i$ be <independent and identically distributed random variables> with <moment-generating function> finite on an open interval about $0$. Put
$$
\Lambda(\theta)=\log\mathbb E e^{\theta Y_1},\qquad I(x)=\sup_{\theta\in\mathbb R}\{\theta x-\Lambda(\theta)\},
$$
allowing $\Lambda=+\infty$ outside its finite domain. The <Cramér theorem> states that $\bar Y_n=n^{-1}\sum_{i=1}^nY_i$ satisfies a <large deviation principle> with speed $n$ and good <rate function> $I$, the <Legendre transform of a cumulant-generating function>. Explicitly, for every closed set $F$ and open set $O$,
$$
\limsup_{n\to\infty}\frac1n\log\mathbb P(\bar Y_n\in F)\leq-\inf_{x\in F}I(x),\qquad
\liminf_{n\to\infty}\frac1n\log\mathbb P(\bar Y_n\in O)\geq-\inf_{x\in O}I(x).
$$
Here $\log0=-\infty$ and $\inf\varnothing=+\infty$. The <rate function> is lower semicontinuous and its finite sublevel sets are compact. \b[The exponential-moment hypothesis is essential for this form of the <Cramér theorem>.]
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