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

By the causality root criterion for an autoregressive model, choose $r>1$ smaller than the modulus of every root of $\Phi$. The function $\Psi =1/\Phi$ is an analytic function on and inside $|z|=r$. Writing $M={\max }_{|z|=r}|1/\Phi (z)|$, the Cauchy estimate gives $|{\psi }_j|\le M{r}^{-j}$. For $h\ge 0$, independence of the noise in the infinite moving-average representation givesThe autocovariance is symmetric in the lag. Thus exponential decay holds withThis is exponential autocovariance decay of a causal autoregression. If $\Phi$ is constant, there are no roots and any $r>1$ works.