Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 26G / a

Lévy's convergence theorem says that probability measures ${\mu }_n$ on ${\mathbb{R}}^d$ converge weakly to $\mu$ if and only if their characteristic functions converge pointwise to the characteristic function of $\mu$. More generally, if the characteristic functions converge pointwise to a function continuous at zero, that function is the characteristic function of a probability measure and the corresponding weak convergence holds.

If $X_n\Rightarrow X$, the continuous mapping theorem givesfor every $u$. Conversely, the assumed one-dimensional convergence at $t=1$ givesfor every $u\in {\mathbb{R}}^d$. Lévy's convergence theorem then yields $X_n\Rightarrow X$. This is the Cramer-Wold theorem.