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

The sequence $X_n$ converges in distribution to $X$ when $\mathbb{E}f(X_n)\to \mathbb{E}f(X)$ for every bounded continuous function $f$.

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 $X_n+Y_n\stackrel{d}{\to }X+Y$.

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.

A family $({\mu }_n)$ of probability measures on a metric space is tight when for every $\eta >0$ there is a compact set $K$ such that ${\sup }_n{\mu }_n(K^c)<\eta$.

Choose $0<\alpha <\epsilon /p$. At dyadic level $m$, Markov inequality and a union bound giveuniformly in $n$. Summing over $m$ shows that, outside a set of probability at most $C_0{C}^{-p}$, all dyadic increments obey this bound. Chaining dyadic approximations and using continuity givesfor all $s,t$. Since $X_0^n=0$, the paths then lie in the stated compact Hölder set by the Arzela-Ascoli theorem. Taking $C$ large proves tightness.