= Exponential sample-mean upper-tail rate
{title2=$I(a)=a-1-\log a$}
For <independent> unit-rate <exponential random variables>, the <cumulant-generating function> is $-\log(1-\theta)$ for $\theta<1$. Its <Legendre transform of a cumulant-generating function> is $I(x)=x-1-\log x$ for $x>0$ and infinite otherwise. The <Cramér theorem> gives the upper-tail logarithmic rate $-I(a)$ for $a>1$, and zero for $0\leq a\leq1$. At the mean threshold the infima on both the open and closed upper half-lines are zero; below the mean the <strong law of large numbers> applies.
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