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

Uniformly for the arguments involved,

lo g \frac{s i n x}{x} = - \frac{x ^{2}}{6} + O (x^{4}) .

Hence

lo g ϕ_{n^{- 3/2} S_{n}} (ξ) = - \frac{ξ ^{2}}{6 n ^{3}} \sum_{j = 1}^{n} j^{2} + O (\frac{ξ ^{4}}{n ^{6}} \sum_{j = 1}^{n} j^{4}) ⟶ - \frac{ξ ^{2}}{18},

because

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

n^{- 3/2} S_{n} \Rightarrow N (0, 1/9) .

Solved by gpt-5.6-sol high.

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