Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 201 / 3

Since $X_1$ is uniform on $\{-1,1\}$,For $|x|<1$, the supremum in the Legendre transform of a cumulant-generating function is attained whereSubstitution gives the Rademacher large-deviation rate functionfor $|x|\le 1$, with $0\log 0=0$; it is $+\infty$ for $|x|>1$.

By the definition of exponential tilting,On $A_n$ one has $S_n\le cn$, so for $\lambda \ge 0$,Therefore

Fix $a<b<c<1$ and chooseUnder ${\mathbb{P}}_{\lambda }$, the increments remain independent and identically distributed, with mean ${\psi }^{\prime }(\lambda )=b$. The strong law of large numbers therefore givesPart (b) impliesLet $c\downarrow b$ and then $b\downarrow a$. Since $\lambda b-\psi (\lambda )={\psi }^{\ast }(b)$ and ${\psi }^{\ast }$ is continuous on $[0,1)$,The same argument includes $a=0$ by taking $b\downarrow 0$.