Linear process (time series, $X_t=\mu +{\sum }_j{\psi }_j{\epsilon }_{t-j}$)

A linear process is an L2-convergent white-noise filter $X_t=\mu +{\sum }_{j\in \mathbb{Z}}{\psi }_j{\epsilon }_{t-j}$, with square-summable coefficients. It is weakly stationary, and its autocovariance is ${\sigma }_{\epsilon }^2{\sum }_j{\psi }_j{\psi }_{j+h}$. A causal representation restricts the coefficients to $j\ge 0$; it is then an infinite moving-average representation.

When the transfer function is $(1+\theta z)/((1-rz)(1-sz))$ with distinct $r,s$ of modulus less than one, partial fractions give coefficients ${\psi }_j=a{r}^j+d{s}^j$, where $a+d=1$ and $-as-dr=\theta$. White-noise orthogonality gives covariance at lag $h\ge 0$ as ${\sigma }^2[a^2{r}^h/(1-r^2)+d^2{s}^h/(1-s^2)+ad(r^h+s^h)/(1-rs)]$. This combines stable recursion with a closed geometric-sum covariance.