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

Lévy's continuity theorem states that if characteristic functions ${\varphi }_n$ converge pointwise to a function $\varphi$ continuous at zero, then $\varphi$ is a characteristic function and the corresponding laws converge weakly to its law. Conversely, convergence in distribution implies pointwise convergence of characteristic functions.