Linear Process

Definition

A linear process $\{X_t\}$ is a weighted linear combination of white noise terms:

$$X_t = \sum_{j=-\infty}^{\infty} \psi_j Z_{t-j}$$

where $\{Z_t\} \sim \text{WN}(0, \sigma^2)$ and $\{\psi_j\}$ is a sequence of constants with $\sum_{j=-\infty}^{\infty} |\psi_j| < \infty$ (absolute summability, guarantees convergence).

Properties

  • $E[X_t] = 0$
  • $\text{Var}(X_t) = \sigma^2 \sum_{j=-\infty}^{\infty} \psi_j^2$
  • $\gamma(h) = \sigma^2 \sum_{j=-\infty}^{\infty} \psi_j \psi_{j+h}$

All three are independent of $t$ → the linear process is stationary.