 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-3/26g/a/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

X_{n} \Rightarrow X

⟨ u, X_{n} ⟩ \Rightarrow ⟨ u, X ⟩

for every

u

. Conversely, the assumed one-dimensional convergence at

t = 1

gives

φ_{X_{n}} (u) = E e^{i ⟨ u, X_{n} ⟩} ⟶ E e^{i ⟨ u, X ⟩} = φ_{X} (u)

for every

u \in R^{d}

. Lévy's convergence theorem then yields

X_{n} \Rightarrow X

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