Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 208 / 2 / b / Solution

Since $\mathbb{E}f(X)=0$ and the moment-generating function is finite near zero, $\log F(u)=O(u^2)$. Consequentlywhich proves $F(\lambda /2^m{)}^{2^m}\to 1$.

For ${\lambda }_{\ast }=C_P(X{)}^{-1/2}$, part (a) and the elementary bound $-\log (1-x)\le 4x/3$ for $0\le x\le 1/4$ giveThus $F({\lambda }_{\ast })\le 3$.