Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 27H / c / Solution

Uniformly for the arguments involved,Hencebecause $n^{-3}\sum j^2\to 1/3$ and the error is $O(n^{-1})$. The limiting characteristic function is that of $N(0,1/9)$. Lévy's theorem gives