 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-4/26k/b/iii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

\sum_{k = 1}^{n} a_{k} Y_{k} \sim N (0, v_{n}), v_{n} = \sum_{k = 1}^{n} a_{k}^{2} .

Since

v_{n} \to v := \sum_{k \geq 1} a_{k}^{2} < \infty

exp (- \frac{1}{2} v_{n} t^{2}) ⟶ exp (- \frac{1}{2} v t^{2}) .

N (0, k \geq 1 \sum a_{k}^{2}) .

On the given common probability space, the same variance calculation actually proves convergence in

L^{2}

as well.

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