Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 2 / 29J (Optimization and Control)

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

Use the usual state-space assumption that the prior $X_0$ is independent of future process and observation noises. Suppose inductively $X_t\mid {\mathcal{Y}}_t\sim N({\hat{X}}_t,V_t)$. Prediction through the linear dynamics gives $X_{t+1}\mid {\mathcal{Y}}_t\sim N(A{\hat{X}}_t,M_t)$, where $M_t=A{V}_t{A}^T+{\Sigma }_{\epsilon }$. The pair $(X_{t+1},Y_{t+1})$ is conditionally multivariate normal, with means $A{\hat{X}}_t,CA{\hat{X}}_t$, cross covariance $M_t{C}^T$, and observation covariance $R_t={\Sigma }_{\eta }+C{M}_t{C}^T$. Applying part (a) gives the Kalman filterThis proves the inductive Gaussian assertion. The inverse requires nonsingular innovation covariance, as in the printed formula. Importantly, prediction uses $A{\hat{X}}_t$: the TeX conversion drops this $A$ in one place, whereas the original PDF includes it.

Write ${\sigma }^2={\Sigma }_{\eta }>0$ and assume $V_0>0$. The scalar Kalman filter reduces toTaking reciprocal variance and dividing the mean by variance givesThusSince the process noise is zero and $A=1$, $X_t=X_0$ and $Y_j=X_0+{\eta }_j$. The strong law of large numbers gives $t^{-1}{\sum }_j{Y}_j\to X_0$ almost surely. The prior's finite weight vanishes in the displayed formula, so both limits equal $X_0$ almost surely. A zero prior variance instead means $X_0$ is known, and the assertion follows directly without reciprocal-variance formulas.