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.

Solution

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

 Solution (source code)