Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 208 / 1 / 1 / 6

Use the original-noise coefficients ${\psi }_0=1$ and ${\psi }_j=-11\cdot 4^{-j}$ for $j\ge 1$. The autocovariance function of a causal linear process is ${\gamma }_X(h)={\sum }_{j\ge 0}{\psi }_j{\psi }_{j+|h|}$, since the original noise has variance $1$. ThusFor $h\ge 1$,By covariance symmetry,In particular, all nonzero-lag covariances are negative, despite the positive autoregressive coefficient.