Weak Stationarity

Definition (★ 考试重点)

$\{X_t\}$ is (weakly) stationary if:

  1. $E(X_t) = \mu$ for all $t$ (constant mean)
  2. $\text{Var}(X_t)$ independent of $t$ (constant variance)
  3. $\text{Cov}(X_r, X_s) = \text{Cov}(X_{r+h}, X_{s+h})$ for all $r, s, h$ (covariance depends only on lag)

Verification Strategy

  1. Compute $E(X_t)$ — depends on $t$? → not stationary, stop.
  2. Compute $\text{Var}(X_t)$ — depends on $t$? → not stationary, stop.
  3. Compute $\text{Cov}(X_t, X_{t+h})$ — depends on $t$? → not stationary.

All free of $t$ → weakly stationary.