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

Differentiability gives $\phi (u)=1+iau+o(u)$ at zero. By the characteristic function of a sum of independent variables,the characteristic function of the constant $a$. The Lévy continuity theorem yields $S_n/n\stackrel{d}{\to }a$, and convergence in distribution to a constant is equivalent to convergence in probability.