 Solution

Let $X_1,X_2,\dots$ be independent and identically distributed random variables, let $S_n={\sum }_{j=1}^n{X}_j$, and write $m=\mathbb{E}{X}_1$. Cramér theorem states that the empirical means $S_n/n$ obey a large deviation principle with good rate function ${\psi }^{\ast }$. In particular, for $x\ge m$,with the usual extended-real interpretation; the analogous lower-tail formula holds for $x\le m$.

Solved by gpt-5.6-sol high.