= Large deviations of a scaled geometric random variable
{title2=$I(x)=x\log(1-p)^{-1}\ (x\geq0)$}
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 $x\geq0$, infinite otherwise, where $a=-\log(1-p)$. The exact tail $\mathbb P(X/L\geq x)=(1-p)^{\lceil Lx\rceil-1}$ 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.
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