 Solution

Let $Y_i$ be independent and identically distributed random variables with moment-generating function finite on an open interval about $0$. Putallowing $\Lambda =+\infty$ outside its finite domain. The Cramér theorem states that ${\bar{Y}}_n=n^{-1}{\sum }_{i=1}^n{Y}_i$ 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 $O$,Here $\log 0=-\infty$ and $\inf \varnothing =+\infty$. The rate function is lower semicontinuous and its finite sublevel sets are compact. The exponential-moment hypothesis is essential for this form of the Cramér theorem.