Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 37 / 2 / e / i / Solution

The covariance is zero. Let ${\mathcal{F}}_t=\sigma ({\epsilon }_s:s\le t)$. The stationary noise-series representation makes $f(X_t)$ measurable with respect to ${\mathcal{F}}_t$, whereas the martingale difference sequence property gives $\mathbb{E}[X_{t+h}\mid {\mathcal{F}}_{t+h-1}]=0$. The law of total expectation therefore givesSince $\mathbb{E}{X}_{t+h}=0$, this is the required covariance. All products are integrable by the Cauchy-Schwarz inequality, using finite second moments of $X$ and the stipulated square integrability of $f(X_t)$.