Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 37 / 1 / a / iii / Solution

A noncausal stationary autoregression exists. On the two-sided time axis defineThis series converges in the sense of mean-square convergence, and direct subtraction gives $X_t-2X_{t-1}={\epsilon }_t$. The expected value and autocovariance areConsequently this is a weakly stationary process and, by Gaussianity, a strictly stationary process. It is an example of noncausal stationary autoregression, since $X_t$ uses future noise. In particular, $X_{t-1}$ is correlated with ${\epsilon }_t$, so the usual causal variance recursion is inapplicable. If an additional assumption required the driving noise to be independent of past observations, this solution would be excluded; that assumption is not stated here.