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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-218/2/d/i/solution