Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 209 / 2 / c / Solution

Fix $u$. Put $m=\phi (u)$ and form the centered real vectorsThey are bounded independent and identically distributed random variables, so the multivariate central limit theorem applies to $n^{-1/2}{\sum }_j{Y}_j=(\mathrm{Re}{\mathcal{C}}_n(u),\mathrm{Im}{\mathcal{C}}_n(u){)}^T$.

To identify its limiting covariance matrix, put $Z=e^{iu{X}_1}-m$. Its complex covariance and pseudo-covariance are respectivelyFor $Z=A+iB$, $|Z{|}^2=A^2+B^2$ and $Z^2=A^2-B^2+2iAB$. ThusThe two conditions on $\Gamma (u)$ identify exactly this real covariance matrix: the second is ${\mathrm{Cov}}_{\mathbb{C}}(\Gamma (u),\overline{\Gamma (u)})=\mathbb{E}\Gamma (u{)}^2$, with a conjugate in its second argument. A centered multivariate normal distribution is determined by its covariance matrix, including when that matrix is singular. ThereforeThis proves convergence in distribution at each fixed $u$; it does not assert convergence of the entire empirical characteristic process.