Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 5 / b / Solution

Independence makes the joint law of $(X_n,Y_n)$ the product of its marginal laws. Weak convergence of both marginals implies convergence of these product measures to the product law of independent copies $(X,Y)$. The addition map is continuous, so the continuous mapping theorem gives $X_n+Y_n\stackrel{d}{\to }X+Y$.