 Large deviations of a scaled geometric random variable

For a geometric random variable $X$ with $\mathbb{P}(X=k)=p(1-p{)}^{k-1}$, $0<p<1$, the family $X/L$ has large-deviation speed $L$ and good rate function $I(x)=ax$ for $x\ge 0$, infinite otherwise, where $a=-\log (1-p)$. The exact tail $\mathbb{P}(X/L\ge x)=(1-p{)}^{\lceil Lx\rceil -1}$ gives the upper bound. A lattice point tending to any positive $x$ supplies the open-neighborhood lower bound; neighborhoods of zero have probability tending to one. Finite sublevel sets are compact intervals.