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For the unit-rate exponential distribution, direct integration givesFor $x>0$, maximizing $\theta x+\log (1-\theta )$ gives $\theta =1-1/x$, and therefore $I(x)=x-1-\log x$. For $x\le 0$ the supremum is infinite, as seen by taking $\theta \to -\infty$.

For $a>1$, $I$ is increasing on $[a,\infty )$, and its continuity gives ${\inf }_{[a,\infty )}I={\inf }_{(a,\infty )}I=I(a)$. The closed-set upper bound and open-set lower bound of the Cramér theorem squeeze the desired limit to $-I(a)$. For $a=1$, both infima are $0$, because $I(1)=0$ and $I(x)\to 0$ as $x\downarrow 1$. For $0\le a<1$, the strong law of large numbers gives $S_n/n\to 1$ almost surely, so the tail probability tends to $1$; at $a=0$ it is identically $1$.

Consequently the exponential sample-mean upper-tail rate isThe threshold case uses the rate-function lower bound on the open half-line, rather than applying the law of large numbers at equality.