Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 2 / 29J / a / Solution

Assume $V_{YY}$ is invertible. PutThis and $Y$ are jointly multivariate normal, with $\mathrm{Cov}(R,Y)=0$ and $\mathrm{Cov}(R)=V_{XX}-V_{XY}{V}_{YY}^{-1}{V}_{YX}$. Their joint Gaussian characteristic function factors when the cross covariance vanishes, so they are independent. Conditioning on $Y=y$ therefore leaves the law of $R$ unchanged, proving