Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 201 / 3 / a

Let $\mu =\mathbb{E}{X}_1$ and let the cumulant-generating function and its Legendre transform beUnder the permitted assumption that $\psi$ is finite everywhere, the right-tail form of the Cramér theorem is, for $a\ge \mu$,Use $\log 0=-\infty$. The restriction to nonnegative parameters is valid on this side of the mean: convexity gives $\psi (\theta )\ge \mu \theta$, so negative parameters cannot improve the value zero available at $\theta =0$. For $a<\mu$, the right-tail probability tends to one by the strong law of large numbers, so its logarithmic rate is zero; it is not generally $-I(a)$.

The full large deviation principle bounds probabilities of closed sets from above and open sets from below with rate $I$. For the displayed tail equality, a possible finite upper endpoint also needs attention. If $b=\mathrm{esssup}{X}_1<\infty$, then at $a=b$ the event requires every summand to equal $b$; its probability is $\mathbb{P}(X_1=b{)}^n$, and $I(b)=-\log \mathbb{P}(X_1=b)$. The identity follows by taking $\theta \to \infty$ in $\mathbb{E}{e}^{\theta (X_1-b)}$. Above $b$, both sides have infinite rate. The theorem describes exponential decay, including endpoints and impossible events.