The <autoregressive process of order one> is causal exactly when
$$
|\phi|<1,
$$
because then $X_t=\sum_{j\geq0}\phi^j\varepsilon_{t-j}$ converges in mean square. Its autocovariance is
$$
\gamma_X(h)=\frac{\sigma^2}{1-\phi^2}\phi^{|h|},
\qquad h\in\mathbb Z.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 218 / 2 / d / i / Solution

{title2=$\operatorname{AR}(1)$}

An autoregressive process of order one satisfies $X_t=\phi X_{t-1}+\varepsilon_t$. It is causal and weakly stationary when $|\phi|<1$.

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